2 Further main results

We say that the set $P=X+{\tfrac 12}B_d$ is a sphere packing of $\mathbb {R}^d$ (associated to a set $X\subset \mathbb {R}^d$ ) if $|x-y|\in \{0\}\cup [1,\infty )$ for all $x,y\in X$ , where $B_d:=\{y\in \mathbb {R}^d : |y|\leq 1\}$ is the unit ball and $|\cdot |$ is the Euclidean norm. For a sphere packing P, we define its density by

$$ \begin{align*} \mathrm{dens}(P):=\displaystyle\limsup_{t\rightarrow+\infty} \displaystyle \displaystyle\frac{\mathrm{vol}(P\cap tQ_d)}{\mathrm{vol}(tQ_d)}, \end{align*} $$

where $Q_d:=\left [-\frac {1}{2},\frac {1}{2}\right ]^d$ is the unit cube. The setup is as follows.

For now on, we will stick with the formulation using K rather than A (prescribing rather than forbidding), as it fits better our scheme of results and constructions.

We now introduce some known facts about lattices. A (full rank) lattice $\Lambda \subset \mathbb {R}^d$ is a discrete subgroup of $(\mathbb {R}^d,+)$ that contains d linear independent vectors. We let

$$ \begin{align*}\ell(\Lambda):=\{|\lambda| : \lambda\in \Lambda\setminus\{0\}\} \end{align*} $$

denote the lengths of $\Lambda $ and $\min \ell (\Lambda )$ denote its minimal norm. Given a lattice $\Lambda $ , one can associate a sphere packing

$$ \begin{align*}P_\Lambda := \frac{1}{r}\Lambda + {\tfrac12}B_d \quad (\text{with } \ r=\min \ell(\Lambda)) \end{align*} $$

and show that

$$ \begin{align*}\mathrm{dens}(\Lambda):=\mathrm{dens}(P_\Lambda)=\frac{\mathrm{vol}(\tfrac{r}{2}B_d)}{\mathrm{ vol}(\mathbb{R}^d/\Lambda)}.\end{align*} $$

We say that a lattice $\Lambda $ is K-admissible if the packing $P_\Lambda $ above is K-admissible – that is, if

$$ \begin{align*}\frac{\ell(\Lambda)}{\min \ell(\Lambda)} \subset K \cup (\sup(K),\infty). \end{align*} $$

An even unimodular lattice $\Lambda $ is one such that $\mathrm {vol}(\mathbb {R}^d/\Lambda )=1$ and $\ell (\Lambda )\subset \{\sqrt {2n} : n\geq 1\}$ (such lattices are integral and self-dual). These lattices have been widely studied and classified in the literature. It is known that they can only exist in dimensions multiple of $8$ and, due to a classical theorem of Voronoi, that there are only finitely many of them in each dimension (modulo symmetries). It is known that (see [Reference Conway and Sloane12, p. 194, Cor. 21])

$$ \begin{align*}(\min \ell(\Lambda))^2 \leq a_d := 2\bigg\lfloor \frac{d}{24} \bigg\rfloor +2. \end{align*} $$

An even unimodular lattice attaining the above bound is called extremal. The $E_8$ , $E_8^2, D_{16}^+$ and Leech lattices are the only even unimodular extremal lattices up to dimension $24$ . In dimensions $32$ and $40$ , there are more than $10^7$ and $10^{51}$ of such lattices, respectively; however, in dimension $48$ , there are (so far) only $4$ known lattices: $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ (see Nebe [Reference Nebe22]). Moreover, it is known that extremal lattices cannot exist in sufficiently large dimensions [Reference Mallows, Odlyzko and Sloane20] (as modular forms with several vanishing Fourier coefficients and very large weight necessarily have negative coefficients). The current best bound is due to Jenkins and Rouse [Reference Jenkins and Rouse19], and it states that

$$ \begin{align*}d_{\max} := {\sup}\{\text{rank}(\Lambda) : \Lambda \text{ is an even unimodular extremal lattice} \}\leq 163264. \end{align*} $$

Indeed, one can show that for any $\beta>0$ , there exists D such that there is no even unimodular lattice of rank $d> D$ and minimal squared norm larger than $a_{d}-\beta $ . For more information on extremal lattices, see [Reference Scharlau and Schulze-Pillot26].

We now state three other main results of this paper. The first is an analogue of the Cohn and Elkies linear programming bound for $\Delta _d(K)$ .

Theorem 3. Let $K\subset [1,+\infty )$ be bounded and such that $1\in K$ . Define

$$ \begin{align*}\Delta_d^{\mathrm{LP}}(K):=\mathrm{vol} \left( {\tfrac12}B_d\right)\inf\displaystyle\frac{F(0)}{\widehat{F}(0)},\end{align*} $$

where the infimum is taken over all nonzero functions $F\in L^1(\mathbb {R}^d) \cap C(\mathbb {R}^d)$ such that

$$ \begin{align*} F(x) \leq 0 \ \text{ for }\ |x|\in K\cup(\sup(K),\infty) \quad \text{and} \quad \widehat{F}(x) \geq 0 \text{ for all } x. \end{align*} $$

Then

$$ \begin{align*} \Delta_d(K)\leq\Delta_d^{\mathrm{LP}}(K). \end{align*} $$

Theorem 4. Let $8\leq d\leq 1200$ , where d is divisible by $8$ but . Define $l_d$ as in (1) and

$$ \begin{align*}K_d = \frac{1}{\sqrt{a_d}}\{\sqrt{a_d},\sqrt{a_d+2},\sqrt{a_d+4},\ldots ,\sqrt{l_d}\}. \end{align*} $$

Also let $c_d$ be given by Table 1 ( $c_d=0$ if $d=8,24, 48$ ). Then there exists a nonzero radial function $H:\mathbb {R}^d\to \mathbb {R}$ of Schwartz class such that

• • $H(x)\leq 0$ if $|x|^2>l_d$ ;

• • $\widehat H(x)\geq 0$ if $|x|> c_d$ ;

• • $H(x)=\widehat H(x)=0$ if $|x|^2 \in \{a_d,a_d+2,\ldots \}$ ;

• • $\{|x|^2 : H(x)=0 \text { and } |x|^2> l_d^-\}=\{l_d,l_d+2,\ldots \}$ ;

• • $\{|x|^2 : \widehat H(x)=0 \text { and } |x|^2>c_d\}=\{a_d,a_d+2,\ldots \}.$

Moreover, if $d=48$ we additionally have that $\{|x|^2 : H(x) < 0\} \cap (0,10) =(6,8)$ .

Table 1 Values of $c_d$ for $d=8,16,24,\ldots ,1200$ . One should read it left to right top to bottom. From dimension $d=536$ onwards, computational time was too high, and we simply took $c_d=a_d-2$ , which can be verified much faster. For $d<536$ , the numbers ${c_d}$ give a good rational approximation of the last sign change of the function $s\mapsto \widehat H(\sqrt {s})$ from Theorem 4. These numbers can also be found in the ancillary file cnumbers on the arXiv submission of this paper (arXiv.org:2308.03925).

In the theorem above, $|x|^2>l_d^-$ means that $|x|^2>l_d-\epsilon _d$ for some small $\epsilon _d>0$ . We note that one can indeed build functions H for all dimensions congruent to $16$ modulo $24$ using the same techniques of Theorem 4. However, it turns out that $\widehat H(x)\leq 0$ for $|x|>o(a_d)$ (numerically), although $H(x)\leq 0$ for $|x|^2>l_d$ and $H(0)=\widehat H(0)>0$ . One should also notice that the numbers $c_d$ seem to satisfy (for small d)

$$ \begin{align*}c_d=a_d-2-O(1) \ \text{ if } d\equiv 8 \text{ mod } 24 \quad \text{and} \quad c_d= a_d-6-O(1) \ \text{ if }d\equiv 0 \text{ mod } 24. \end{align*} $$

Also, in fact, $c_d$ is an approximation from the right of the last simple root of $\widehat H(x)$ . All these facts give a heuristic explanation why we only get results free from spectral conditions in dimensions $8,24$ and $48$ (hence a result for all sphere packings, periodic or not). It goes as follows: Experimentally, the $O(1)$ in $c_d$ is less than $1$ for small d and $c_d$ increases with d on each equivalence class modulo $24$ , which means that if $d\geq 72$ , then $a_d\geq 8$ , and so $c_d\geq 1$ . Thus, $\widehat H$ would never be nonnegative. For $d=8,24,48$ , we have $c_d=0-O(1)<0$ . Thus, $\widehat H\geq 0$ (see Figure 1). For the remaining small dimensions not equal to $16$ modulo $24$ , which are, $d=32$ and $d=56$ , we have $c_{32}=2-O(1)$ and $c_{56}=4-O(1)$ , which are positive, and so $\widehat H$ is not nonnegative.

Theorem 5. Let d, $a_d$ , $K_d$ and $c_d$ be as in Theorem 4. Let $P=\Lambda +Y+{\tfrac 12}B_d$ be some $K_{d}$ -admissible periodic sphere packing such that $\min \ell (\Lambda ^*)> \sqrt {c_d}$ . Then

$$ \begin{align*}\mathrm{dens}(P)\leq \mathrm{vol} \left( B_d\right) \left( \frac{\sqrt{a_d}}{2}\right)^d.\end{align*} $$

In case $\#Y=1$ , equality above occurs if and only if $\sqrt {a_d}\Lambda $ is an even unimodular extremal lattice. We conclude that if $d\in \{8,24,48\}$ , then

$$ \begin{align*}\Delta_d(K_d) = \Delta_d^{\mathrm{LP}}(K_d) = \mathrm{vol} \left( B_d\right) \left( \frac{\sqrt{a_d}}{2}\right)^d. \end{align*} $$

Figure 1 This is a plot of the functions $s\mapsto \widehat H(\sqrt {s})e^{\pi s}$ for $d=8$ (black), $d=24$ (blue) and $d=48$ (red), normalized so $\widehat H(0)=1$ .

Figure 2 This is a plot of the functions $s\mapsto \widehat H(\sqrt {s})e^{\pi s}$ for $d=72$ (red) and $d=80$ (blue), normalized so $\widehat H(0)=1$ . For $d=80$ , we have multiplied the function by $(s+1)^2$ for aesthetic reasons.

Note that if $d\in \{8,24,48\}$ , then $c_d=0$ , and this shows that even unimodular extremal lattices maximize density for the among all $K_d$ -admissible sphere packings (periodic or not). Since $K_8=K_{24}=\{1\}$ , the packing problem in these dimensions is unconstrained, and the $E_8$ and Leech lattices are the only extremal lattices in these dimensions, this shows they have maximal density. The above theorem puts the results of [Reference Cohn, Kumar, Miller, Radchenko and Viazovska10, Reference Viazovska27] in a larger family of packing problems that can be solved by linear programming methods and construction via modular forms. The fact that for $d=48$ we have $H(x) < 0$ for $6<|x|^2 < 8$ , allows us to enlarge $K_{48}$ to $\frac 1{\sqrt {6}}([\sqrt {6},\sqrt {8}]\cup {\sqrt {10}})$ and deduce Theorem 1. The same enlargement is possible in every dimension d; that is, we could fill $K_d$ between a couple of its points and prove a slightly stronger result. However, for simplicity, we left the statement as it is.

One could ask if it is possible to extend Theorems 4 and 5 to all dimensions $d\leq 163264$ . For this, one would need to greatly optimize the numerical procedure we explain in the proof of Theorem 4, and use specialized software and several days of running time to increase $1200$ to something of the order of $10000$ . Rough experimental estimations show that the running time we have for the proof of Theorem 4 is roughly $O(1.1^{d/8})$ -secs, so it seems the complexity of our algorithm is increasing exponentially. Even if one manages to reduce this ratio to (being generous) $1.001$ , reaching $d\approx 170000$ seems unreasonable.

One could also ask if we could prove Theorem 5 with no assumption on the minimal norm on $\Lambda ^*$ . That might be possible, but we believe it to be impossible via the linear programming approach that we use with the same set $K_d$ . The functions H computed in Theorem 4 are in a way unique, and one could actually show they are by extending the interpolation formulas of [Reference Cohn, Kumar, Miller, Radchenko and Viazovska11] to all dimensions multiple of $8$ . The issue is that the functions $\widehat H(x)$ of Theorem 4 do have a simple zero very near $|x|^2=c_d$ and so have negative values in the region $0<|x|^2<c_d$ . However, it might be possible to remove the condition on $\Lambda ^*$ by enlarging $l_d$ and finding the corresponding ‘magic’ functions. We have tried this approach in small dimensions, replacing $l_d$ by $l_d+\delta $ , for some small even $\delta>0$ , although unsuccessfully. It might be the case that $\delta $ needs to be very large; however, that greatly complicates the modular form constructions. We leave this question for future work. Nevertheless, we expect this $\delta $ to exist because when $\delta =\infty $ , the only $K_d$ -admissible lattices with minimal norm $\sqrt {a_d}$ are integral even lattices, and such lattices are less dense than extremal ones.

Conjecture 3. Let $\Lambda \subset \mathbb {R}^d$ be an even unimodular lattice with minimal norm $\sqrt {a}$ , for some even integer a. Then for some even $l>a$ , we have that $\Lambda $ has maximal density among any $\frac 1{\sqrt {a}}\{\sqrt {a},\sqrt {a+2},\ldots ,\sqrt {l}\}$ -admissible sphere packing; that is,

$$ \begin{align*}\Delta_d\left(\frac1{\sqrt{a}}\{\sqrt{a},\sqrt{a+2},\ldots ,\sqrt{l}\}\right) = \mathrm{dens}(\Lambda). \end{align*} $$

If this conjecture is true in dimension d, then for $2\leq a \leq a_d$ , one could define $L(a,d)=l$ , where l is the smallest such that Conjecture 3 is true. We already know that $L(2,8)=2, L(4,24)=4$ , and we have shown that $L(6,48)\leq 10$ . We believe (and it is somewhat believed in the community) that $L(6,48)=6$ and that extremal lattices maximize density with no constraints in dimension $48$ . It would be also interesting to find $L(2,16)$ and $L(2,24)$ and show that the lattices $E_8^2, D_{16}^+$ and all the $24$ -dimensional Niemeier lattices with root are optimal.

Another curious question is: Is there a finite K such that $\mathbb {Z}^2$ is K-admissible and with maximal density?Footnote ³ If so, how small $\#K$ can be?

2.1 One-dimensional sphere packings

Unconstrained one-dimensional sphere packings are trivial to construct as unit intervals tile the line. However, finding optimal one-dimensional K-admissible sphere packings for an arbitrary given set K seems to be a difficult question. Here, we are concerned with periodicity: When can we make sure that there exists some optimal one-dimensional packing which is periodic? Unfortunately, greedy choice usually does not give an optimal construction. By greedy choice, we mean one starts with some configuration $\cup _{i=1}^N (a_i+I)$ for $a_1<a_2<\ldots <a_{N}$ (with $I=[0,1]$ ) and then takes an interval $a_{N+1}+I$ with $a_{N+1}\geq 1+a_N$ as small as possible so that $\cup _{i=1}^{N+1} (a_i+I)$ still is K-admissible. For example, first consider the case where $K=\{1,\alpha \}$ for some $\alpha>2$ . Then greedy choice gives the packing P where one puts two unit intervals glued together, and then a gap of length $\alpha -1$ , and then repeats this configuration periodically. One can see this is optimal by noting that, in any given K-admissible packing, the distance between the centers of any unit interval and the second one after it must be at least $1+\alpha $ . This means that an interval of size $N(1+\alpha )$ contains at most $2N$ unit intervals, which shows that $\Delta _1(\{1,\alpha \}) \leq 2/(1+\alpha )$ , and this is attained by P. However, this strategy does not produce the best packing in general. For instance, if $K=\{1,\alpha ,\beta \}$ with $2\alpha \geq \beta>2\alpha -1>\alpha $ , then greedy choice gives the packing $P_\beta =I +(1+\beta )\mathbb {Z}+\{0,1\}$ , which has density $2/(1+\beta )$ . This is not optimal since $P_\alpha =I + \alpha \mathbb {Z} $ has density $1/\alpha>2/(1+\beta )$ . In Lemma 15, we completely solve this problem for all choices of $\alpha $ and $\beta $ .

In Proposition 7, we show that when K is a compact set, then optimal packings for $\Delta _d(K)$ always exist. However, it is not guaranteed that they are periodic, as this is not even known in the unconstrained case. Nevertheless, we expect them to be periodic in the one-dimensional case. The following result proves this in the ‘almost’ finite case.

Theorem 6. Let $K\subset [1,\infty )$ be a compact set such that $1\in K$ . Assume that K has no accumulation points from the left and only finitely many accumulation points from the right. Moreover, let $K'$ be its set of accumulation points and assume that $(K'+K') \cap K'=\emptyset $ . Then there exists a K-admissible periodic sphere packing P of $\mathbb {R}$ such that $\mathrm {dens}(P)=\Delta _1(K)$ .

In particular, optimal periodic K-admissible packings exist whenever K is finite. However, optimal periodic packings will also exist in the (illustrative) case

$$ \begin{align*}K=\{1,\sqrt{2}, e,\pi\} + \{10^{-n}\}_{n\geq 1}. \end{align*} $$

Conjecture 4. Let $K\subset [1,\infty )$ be a compact set such that $1\in K$ . Then there exists a K-admissible periodic sphere packing of $\mathbb {R}$ with maximal density.

A compact set K can be classified by its sequence of derived sets; that is,

$$ \begin{align*}K,K', K", K"',\ldots ,\end{align*} $$

where $S'$ is the set of points $p\in S$ such that $(p-\varepsilon ,p+\varepsilon )\cap S \setminus \{ p\} \neq \emptyset $ for any $\varepsilon>0$ (the accumulation points of S). Theorem 6 solves the above conjecture for the case $K'=\emptyset $ (i.e., K is finite) and deals with the case $K"=\emptyset $ (i.e., $K'$ is finite) under the condition that points only accumulate from the right and no accumulation point is a sum of two others.

We believe that Conjecture 4 could be very hard to prove, perhaps even false, as this is equivalent to (when $\sup (K)\in K$ ) a generalization of Theorem 13 (which is about linear domino tilings) for an infinite compact sets of symbols $\Sigma $ and domino pieces $\mathcal {D}\subset \Sigma ^*\times \Sigma ^*$ .

3 Generalities

In this section, we establish some basic facts about sphere packing with forbidden distances. Throughout this section, $K\subset [1,\infty )$ will always be a bounded set such that $1\in K$ , and the word sphere will be used to denote any $x+{\tfrac 12}B_d$ for some $x\in \mathbb {R}^d$ .

Proof. The proof is exactly the same as for unconstrained sphere packings [Reference Groemer17]. Let $(P_n)_{n\geq 1}$ be a maximizing sequence of sphere packings such that $\mathrm {dens}(P_n)$ is increasing and converges to $\Delta _d(K)$ . By Proposition 8, we can assume that each $P_n$ is ${k_n} \mathbb {Z}^d$ periodic for some integer $k_n>0$ such that $k_n\nearrow \infty $ and that $P_n$ maximizes the number of spheres one can put inside $k_nQ_d$ . Using Hausdorff’s topology for compact sets and a standard Cantor’s diagonal argument, we can assume that $(P_n)_{n\geq 1}$ converges locally to some packing P, which is K-admissible since K is compact. By maximality, the number of spheres of $P_n$ inside $k_mQ_d$ must not be much smaller than that of $P_m$ (the error must be bounded by the surface area of the boundary of $k_mQ_d$ ). We obtain

$$ \begin{align*}\mathrm{vol}(P_n \cap k_m Q_d)/\mathrm{vol}(k_m Q_d)> \mathrm{vol}(P_m \cap k_m Q_d)/\mathrm{vol}(k_m Q_d) + O(1/k_m) = \mathrm{dens}(P_m) + O(1/k_m) \end{align*} $$

for all $n>m$ , where $O(1/k_m)$ comes from the spheres that touch the boundary of $k_mQ_d$ . Taking $n\to \infty $ , we obtain $\mathrm {vol}(P \cap k_m Q_d)/\mathrm {vol}(k_m Q_d) \geq \mathrm {dens}(P_m) + O(1/k_m)$ . Taking $m\to \infty $ , we conclude that $\mathrm {dens}(P)\geq \Delta _d(K)$ , which finishes the proof.

Proposition 8. Any K-admissible sphere packing can be approximated by a periodic one. In particular, if $N_t$ denotes the maximum number of spheres one can put inside $tQ_d$ such that the configuration is $K $ -admissible, then

$$ \begin{align*}\lim_{t\to\infty} \frac{N_t \mathrm{vol}({\tfrac12}B_d)}{\mathrm{vol}(tQ_d)} = \Delta_d(K). \end{align*} $$

Proof. Let $P=X+{\tfrac 12}B_d$ be a K-admissible sphere packing. Then $\widetilde P_t = X\cap tQ_d + (t+\sup (K))\mathbb {Z}^d + {\tfrac 12}B_d$ is K-admissible, periodic and

$$ \begin{align*}\mathrm{dens}(\widetilde P_t)=\frac{\mathrm{vol}(P\cap tQ_d)}{\mathrm{vol}(tQ_d)}(1+O(1/t)). \end{align*} $$

We obtain $\limsup _{t\to \infty } \mathrm {dens}(\widetilde P_t) = \mathrm {dens}(P)$ . Let $\delta _t=\frac {N_t \mathrm {vol}({\tfrac 12}B_d)}{\mathrm {vol}(tQ_d)}$ . The same periodization argument shows that $\limsup _{t\to \infty } \delta _t \leq \Delta _d(K)$ . However, if $P=X+t\mathbb {Z}^d$ is a K-admissible periodic sphere packing, by maximality, we must have that $\#(X \cap tQ_d) \leq N_t$ ; hence,

$$ \begin{align*}\delta_t \geq \mathrm{dens}(P) + O(1/t), \end{align*} $$

where $O(1/t)$ accounts for boundary intersections. We obtain $\liminf _{t\to \infty } \delta _t \geq \Delta _d(K)$ . This finishes the lemma.

Proof. Since K is compact, there is a packing $P=X+{\tfrac 12}B_d$ such that $\mathrm {dens}(P)=\Delta _d(K)$ . Since X is countable, we can write $X=\{x_1,x_2,\ldots \}$ . Define the set

$$ \begin{align*} K_0 =\{\alpha \in K : |x_i-x_j|=\alpha \text{ for some } i < j\}. \end{align*} $$

Define $\widetilde {K}:=K_0\cup \{1,\max (K)\}$ . Then, $\widetilde {K}\subset K$ is a countable subset such that $\max ( K)=\max (\widetilde K)$ , and by construction, P is $\widetilde K$ -admissible. We have

$$ \begin{align*}\Delta_d(\widetilde K)\leq \Delta_d(K) = \mathrm{dens}(P) \leq \Delta_d(\widetilde K). \end{align*} $$

This concludes the proof.

Define

$$ \begin{align*} n_d(K):=\max \{\#X : X\subset \mathbb{R}^d \text{ and } |x-y|\in K \text{ for all } x,y\in X \text{ with } x\neq y\}. \end{align*} $$

Since K is bounded, it is clear that X is finite and any maximal set X (which always exist) can be placed inside a sphere of radii $\sup (K)$ . For instance, if $K=\{1\}$ , then $n_d(K)=d+1$ , and this is realized by the $(d+1)$ -simplex. Let $\mathrm {kiss}_d$ denote the kissing number of $\mathbb {R}^d$ – that is, the largest number of equal size spheres that can touch a central sphere with no overlapping. Then it is easy to see that

$$ \begin{align*}n_d([1,2]) \geq 1+\mathrm{kiss}_d. \end{align*} $$

Trivially, this is attained for $d=1$ . It seems to be the case for $d=2$ and unlikely to be false for $d=3$ . It turns out that the number $n_d(K)$ can be extracted from a constrained packing problem if one sets $K_\lambda = K \cup \{\lambda \}$ and sends $\lambda \to \infty $ .

Proposition 10. Let $K \subset [1,\infty )$ be bounded with $1\in K$ and let $ K_\lambda = K \cup \{\lambda \}. $ Then

$$ \begin{align*}\lim_{\lambda\to\infty} \lambda^d \Delta_d(K_\lambda)=n_d(K)\Delta_d. \end{align*} $$

Proof. First, we claim that $\Delta _d(K_\lambda )\geq \frac {n_d(K)\cdot \Delta _d}{\left (\beta +\lambda \right )^d}$ , where $\beta =\sup {K}$ . In order to do that, we will construct a packing $P_\lambda $ that is $K_\lambda $ -admissible and show that $\mathrm {dens}(P_\lambda )\geq \frac {n_d(K)\cdot \Delta _d}{\left (\beta +\lambda \right )^d}$ . The packing $P_\lambda $ will not necessarily have maximal density; however, it will have a nice structure which makes it easy to estimate its density. Let $Y\subset \beta B_d$ be a maximal cluster of K-admissible points attaining $\#{Y}=n_d(K)$ . Let $\widetilde {P_\lambda }=X_\lambda +\frac {\beta +\lambda }{2} B_d$ be an unconstrained periodic sphere packing (with spheres of diameter $\beta +\lambda $ ) such that $\mathrm {dens}(\widetilde {P_\lambda })>\Delta _d-\varepsilon $ . Define the packing

$$ \begin{align*}P_\lambda = X_\lambda + Y+{\tfrac12}B_d. \end{align*} $$

We claim $P_\lambda $ is $K_\lambda $ admissible. To see this note that if $x_\lambda +y$ and $x_\lambda '+y'$ are two points in $X_\lambda +Y$ , then their distance is $\geq \lambda $ if $x_\lambda \neq x_\lambda '$ . If $x_\lambda =x_\lambda '$ , then their distance is $|y-y'|\in K$ . We obtain

$$ \begin{align*} \mathrm{dens}(P_\lambda) & = \lim_{t\to\infty} \frac{\#((X_\lambda+Y)\cap tQ_d) \mathrm{vol}({\tfrac12}B_d)}{\mathrm{vol}(tQ_d)} \\ & = \lim_{t\to\infty} \frac{\#(X_\lambda\cap (t-2\beta -1)Q_d) n_d(K)\mathrm{vol}(\frac{\beta+\lambda}{2} B_d)}{(\beta+\lambda)^d\mathrm{vol}(tQ_d)} \\ & = \frac{n_d(K)\mathrm{dens}(\widetilde{P_\lambda})}{(\beta+\lambda)^d}> \frac{n_d(K)(\Delta_d-\varepsilon)}{(\beta+\lambda)^d}. \end{align*} $$

Since both $\widetilde {P_\lambda }$ and $P_\lambda $ are periodic, equality between limits above is justified. Letting $\varepsilon \to 0$ proves our claim.

Now we claim that $\Delta _d(K_\lambda )\leq \frac {n_d(K)\cdot \Delta _d}{\lambda ^d}$ . Let $Y+{\tfrac 12}B_d$ be a periodic $K_\lambda $ -admissible sphere packing such that $\mathrm {dens}(Y_\lambda )>\Delta _d(K_\lambda )-\varepsilon $ . We can assume that $\lambda>2\beta $ . Define an equivalence relation in Y by saying that $y_1 \sim y_2$ if $|y_1-y_2|\leq \beta $ . This is an equivalence relation since if $y_1\sim y_2$ and $y_2\sim y_3$ but $|y_1-y_3|>\beta $ , then $|y_1-y_3|\geq \lambda $ , but triangle inequality shows that $|y_1-y_3|\leq 2\beta < \lambda $ , a contradiction. Let $\widetilde Y=\{[y_1],[y_2],\ldots \}$ be these equivalence classes, where the $y_j$ ’s are representatives of each class. Observe that $|y_i-y_j|\geq \lambda $ if $i<j$ , that the set $[y_j]$ has only distances in K and is contained in $y_j + {\beta }B_d$ . Thus, $\#[y_j]\leq n_d(K)$ . We obtain

$$ \begin{align*} \Delta_d(K_\lambda) -\varepsilon & < \lim_{t\to\infty} \frac{\#(Y\cap tQ_d) \mathrm{vol}({\tfrac12}B_d)}{\mathrm{vol}(tQ_d)} \\ & \leq \limsup_{t\to\infty} \frac{ \mathrm{vol}({\tfrac12}B_d) \sum_{ [y_j]\cap tQ_d \neq \emptyset } \#[y_j]}{\mathrm{vol}(tQ_d)} \\ & \leq \frac{n_d(K)}{\lambda^d} \limsup_{t\to\infty} \frac{\#(\{y_1,y_2,\ldots \}\cap (t+2\beta)Q_d) \mathrm{vol}(\tfrac{\lambda}{2}B_d)}{\mathrm{vol}(tQ_d)} \\ & \leq \frac{n_d(K)}{\lambda^d} \Delta_d. \end{align*} $$

Letting $\varepsilon \to 0$ finishes the proof.

Observe that we have actually proven the stronger result

$$ \begin{align*}\frac{n_d(K)\cdot \Delta_d}{\left(\beta+\lambda\right)^d} \leq \Delta_d(K_\lambda) \leq \frac{n_d(K)\cdot \Delta_d}{\lambda^d}. \end{align*} $$

In particular, in conjunction with Theorem 3, we have the new linear program to produce upper bounds for kissing numbers.

Corollary 11. Let $\lambda \geq 4$ . Let $F:\mathbb {R}^d\to \mathbb {R}$ be a continuous $L^1$ -function such that $\widehat F\geq 0$ and $F(x)\leq 0$ if $1<|x|<2$ or if $|x|>\lambda $ . Then

$$ \begin{align*}1+\mathrm{kiss}_d \leq \frac{(2+\lambda)^d}{\Delta_d} \frac{F(0)}{\widehat F(0)} \mathrm{vol}({\tfrac12}B_d). \end{align*} $$

The structure of the best configuration for $d=2$ , $K=[1,2]\cup \{\lambda \}$ and large $\lambda $ will look like Figure 3.

Figure 3 Best configuration for $d=2$ and $K=[1,2]\cup \{4\}$ .

As a proof of concept, we now prove that this linear program is sharp for $d=1$ . In this case, we have the inequality $3\leq (2+\lambda )F(0)/\widehat F(0)$ . Define the following symmetric set

$$ \begin{align*}E_\lambda=\bigcup_{n=-N}^{N}[3n-1/2,3n+1/2] \end{align*} $$

with $N=\lfloor (\lambda -1)/6\rfloor $ . Then it is easy to see that $F_\lambda =\mathrm {1}_{E_\lambda }\star \mathrm {1}_{E_\lambda } $ (where $\star $ is the convolution operator) is positive definite and $F_\lambda (x)=0$ if $|x|>\lambda $ or if $1<|x|<2$ , since $E_\lambda +E_\lambda =\cup _{n=-2N}^{2N}[3n-1,3n+1] $ . Moreover, $F_\lambda (0)/\widehat F_\lambda (0)=1/\mathrm {vol}(E_\lambda )=1/(1+2N) \sim 3/\lambda $ . We conclude that $\lim _{\lambda \to \infty } (2+\lambda )F_\lambda (0)/\widehat F_\lambda (0) = 3$ , proving the bound is sharp.

3.1 Proof of Theorem 3

In view of Proposition 8, it suffices to consider just periodic sphere packings $P=Y+\Lambda +{\tfrac 12}B_d $ . Assume first F is of Schwartz class. We apply Poisson summation to obtain

$$ \begin{align*}(\#Y) F(0) \geq \sum_{y,y'\in Y}\sum_{\lambda \in \Lambda} F(\lambda+y-y') = \frac{1}{\mathrm{vol}(\mathbb{R}^d/\Lambda)}\sum_{\xi \in \Lambda^*} \widehat F(\xi) \left| \sum_{y\in Y} e^{2\pi iy\cdot \xi}\right|^2 \geq \frac{(\#Y)^2\widehat F(0)}{\mathrm{vol}(\mathbb{R}^d/\Lambda)}. \end{align*} $$

Rearranging terms, we deduce that $\mathrm {dens}(P)\leq \mathrm {vol}({\tfrac 12}B_d)F(0)/\widehat F(0)$ . In the unconstrained packing problem one could now, by a standard convolution approximation argument, show the same inequality holds when F is only continuous and $L^1$ . However, this trick does not work for constrained sphere packings, since convolutions destroy K-admissibility of functions. Instead, we give below a direct proof based on a new trick involving the Féjer kernel.

Let A be a full rank matrix such that $\Lambda =A\cdot \mathbb {Z}^d$ and let

$$ \begin{align*}f_\Lambda(x)=\displaystyle\sum_{\lambda\in\Lambda} F(x+\lambda).\end{align*} $$

Then we see that $f_\Lambda $ is $\Lambda -$ periodic, that the summation above converges a.e. and that $f_\Lambda \in L^1(\mathbb {R}^d/\Lambda )$ . Note also that

$$ \begin{align*}f_\Lambda\circ A(x)=\displaystyle\sum_{k\in\mathbb{Z}^d} F\circ A(x+k).\end{align*} $$

Observe that $f_\Lambda \circ A \in L^1(\mathbb {R}^d/\mathbb {Z}^d)$ and $(f_\Lambda \circ A)^\wedge (\alpha )= |\det (A)|^{-1} \widehat F\circ A^{-\top }(\alpha )$ , where $\mathrm {vol}(\mathbb {R}^d/\Lambda )=|\det (A)|$ and we use $\top $ for transpose. For N a positive integer, consider the N-th Féjer kernel

$$ \begin{align*} \mathcal{F}_N(x)=\mathbf{1}_{Q_d}(x)\displaystyle\sum_{\substack{\alpha\in\mathbb{Z}^d\\ |\alpha|_\infty<N}} \displaystyle\prod_{j=1}^{d} \left(1-\displaystyle\frac{|\alpha_j|}{N}\right)e^{2\pi i\alpha\cdot x} = N^d \prod_{j=1}^d \left( \frac{\sin(\pi N x_j)}{N\sin(\pi x_j)} \right)^2 \mathbf{1}_{|x_j|<1/2}. \end{align*} $$

Denote by $\star $ the convolution operator. It is well known that the $\mathcal {F}_N$ is an approximate identity – that is, $\mathcal {F}_N\star G(x)\to G(x)$ as $N\to \infty $ , for any x and any bounded continuous function G. The nice bit is that if G is $\mathbb {Z}^d$ -periodic, then convolution simply multiplies Fourier coefficients. Routine computations show that

$$ \begin{align*}(f_\Lambda\circ A) \star \mathcal{F}_N(x) = \frac{1}{\mathrm{vol}(\mathbb{R}^d/\Lambda)}\sum_{\substack{\alpha\in\mathbb{Z}^d\\ |\alpha|_\infty<N}} \widehat F\circ A^{-\top}(\alpha) \displaystyle\prod_{j=1}^{d} \left(1-\displaystyle\frac{|\alpha_j|}{N}\right) e^{2\pi i \alpha \cdot x} \end{align*} $$

is indeed a trigonometric polynomial. We conclude that

$$ \begin{align*} \sum_{y,y'\in Y} (f_\Lambda\circ A) \star \mathcal{F}_N(A^{-1}(y-y')) \geq \frac{(\#Y)^2}{\mathrm{vol}(\mathbb{R}^d/\Lambda)} \widehat F(0). \end{align*} $$

However, letting $m:=\max _{y\neq y'} {|y-y'|}+\sup (K)+\max _{x\in \mathbb {R}^d/\Lambda } |x|$ , the left-hand side above is equal to

$$ \begin{align*} &\sum_{y,y'\in Y}\displaystyle\int_{Q_d} \mathcal{F}_N(x)\left(f_\Lambda\circ A\right)\left(A^{-1}(y-y')-x\right)dx\\ & = \frac{1}{|\det(A)|}\displaystyle\sum_{y,y'\in Y}\displaystyle\int_{\mathbb{R}^d/\Lambda} \mathcal{F}_N(A^{-1}x)f_\Lambda\left(y-y'- x\right)dx\\ &\leq\displaystyle\frac{1}{\mathrm{vol}(\mathbb{R}^d/\Lambda)}\displaystyle\sum_{y,y'\in Y}\displaystyle\sum_{\substack{\lambda\in\Lambda\\ |\lambda|\leq m}} \displaystyle\int_{\mathbb{R}^d/\Lambda} \mathcal{F}_N(A^{-1}x)F(y-y'-x+\lambda)dx\\ &=\displaystyle\sum_{y,y'\in Y}\displaystyle\sum_{\substack{\lambda\in\Lambda\\ |\lambda|\leq m}} \mathcal{F}_N\star(F\circ A)(A^{-1}(y-y'+\lambda))\\ &\xrightarrow{N\rightarrow\infty} \displaystyle\sum_{y,y'\in Y}\displaystyle\sum_{\substack{\lambda\in\Lambda\\ |\lambda|\leq m}} (F\circ A)(A^{-1}(y-y'+\lambda))\\ &=\displaystyle\sum_{y,y'\in Y}\displaystyle\sum_{\substack{\lambda\in\Lambda\\ |\lambda|\leq m}} F(y-y'+\lambda)\\ &\leq \#Y\cdot F(0). \end{align*} $$

The limit above is justified because we have a finite sum. The theorem follows.

4 One-dimensional packings and dominos

We now focus on the one-dimensional case, and we ask for the existence of periodic packings with the maximal density. Let $I=[0,1]$ be the unit interval. We begin with the observation that we can restrict our attention to packings of the half-line $[0,\infty )$ only; that is,

$$ \begin{align*}\Delta_1(K)=\sup_{P \in \mathcal{P}_1(K) \text{ and } P\subset [0,\infty)} \mathrm{dens}^+(P), \end{align*} $$

where $\mathrm {dens}^+(P)=\limsup _{t\to \infty } \mathrm {vol}(P\cap [0,t])/t$ . Indeed, if $P\subset (-\infty ,\infty )$ is a K-admissible periodic packing, then

$$ \begin{align*}\mathrm{dens}(P) \leq \max\{\mathrm{dens}^+(P\cap \mathbb{R}_+),\mathrm{dens}^+(P \cap \mathbb{R}_-)\}.\end{align*} $$

Now note that for any K-admissible packing of $[0,\infty )$ , we can assume that I starts at $0$ and so the packing can be uniquely described as a sequence

$$ \begin{align*}w=a_{1}a_{2}\ldots a_{k}\ldots , \end{align*} $$

where $a_j\in K\cup (\sup (K),\infty )$ represent the distance between consecutive centers of intervals in the packing. From now on, we assume that $\sup (K)\in K$ , and we set $N:=\lceil \sup (K)\rceil $ . Hence, we can always assume that $a_j\in K$ for all j since otherwise, we can reduce that distance without destroying K-admissibility and obtain a denser packing. For simplicity, we just write for K-admissible packings of $[0,\infty )$ with distances between adjacent centers of intervals drawn from K that start with I touching $0$ .

A word will be a finite ordered sequence of elements of the alphabet $\Sigma =K$ . For instance, $w=a_{1}a_{2}\ldots a_{k}$ is a word of length $\#w=k$ . It will be useful to consider the empty word, denoted by $\varnothing $ , which has no elements and has length $\#\varnothing =0$ . We can also define the norm of w by

$$ \begin{align*}|w|=\sum_{i=1}^k a_i. \end{align*} $$

We denote by $\Sigma ^*$ the set of all finite words. We will now construct a domino set $\mathcal {D}$ by

$$ \begin{align*} \mathcal{D}_K:=\big\{(w,w')\in\Sigma^*\times\Sigma^*: & \#w=\#w'=N \text{ and any subword } {s} \text{ of } ww' \\ & \text{ satisfies } |s| \in K \cup (\sup(K),\infty) \big\}. \end{align*} $$

Thus, $\mathcal {D}_K\subset \Sigma ^* \times \Sigma ^*$ . Here, $w{w'}$ is the concatenated word. A linear domino tiling from $\mathcal {D}_K$ is an infinite word

$$ \begin{align*}\gamma=w_1w_2w_3\ldots \end{align*} $$

where $(w_i,w_{i+1})$ is a domino piece from $\mathcal {D}_K$ . We let $\mathcal {T}(\mathcal {D}_K)$ be the set of domino tiling built this way. Given a tiling $\gamma \in \mathcal {T}(\mathcal {D}_K)$ , we let

$$ \begin{align*}\mathrm{dens}(\gamma) = \limsup_{t\to\infty} \frac{\#(w_1\ldots w_t)}{|w_1\ldots .w_t|}. \end{align*} $$

We say $\gamma $ is periodic if for some n, we have $w_{i+n}=w_i$ for all i.

Proposition 12. There is a canonical bijection between . This bijection maps periodic to periodic and satisfies

$$ \begin{align*}\mathrm{dens}^+(P)=\mathrm{dens}(\gamma_P). \end{align*} $$

In general, we define a linear domino game as a tuple $(\Sigma ,\mathcal {D})$ , where $\Sigma $ is a set of symbols and $\mathcal {D}\subset \Sigma ^*\times \Sigma ^*$ are domino pieces, where $\Sigma ^*$ is the set of all finite words in the alphabet $\Sigma $ . The set of linear domino tilings $\mathcal {T}(\mathcal {D})$ are those infinite words such that $\gamma =w_1w_2w_3\ldots $ , where $(w_i,w_{i+1})\in \mathcal {D}$ for all i. We say that a map $f:\Sigma ^* \to \mathbb {R}_+$ is a norm function when it satisfies the following properties:

1. (a) For any $w_1,w_2,w_3\in \Sigma ^*$ , the function $k \geq 0 \mapsto \frac {\#(w_1w_2^kw_3)}{f(w_1w_2^kw_3)}$ is monotone;

2. (b) The function $k>0\mapsto \frac {\#(w_2^k)}{f(w_2^k)}$ is a positive constant;

3. (c) For any $\epsilon>0$ , there is $\delta>0$ such that if $w'$ is a sub-word of w with $\#w-\#w'\leq \epsilon $ , then $|f(w)-f(w')|\leq \delta $ .

Note that if f is a norm function, then

$$ \begin{align*}\frac{\#(w_1w_2^kw_3)}{f(w_1w_2^kw_3)} = \frac{\#(w_2^k)+O(1)}{f(w_2^k)+O(1)} = \frac{\#(w_2^k)+O(1)}{\frac{f(w_2)}{\#w_2}\#(w_2^k)+O(1)} = \frac{\#w_2}{f(w_2)}+O(1/k) \end{align*} $$

as $k\to \infty $ . For a norm function f and a linear domino tiling $\gamma =w_1w_2w_3\ldots \in \mathcal {T}(\mathcal {D})$ , we let

$$ \begin{align*}\mathrm{dens}_f(\gamma)=\limsup_{t\to\infty} \frac{\#(w_1\ldots .w_t)}{f(w_1\ldots .w_t)}. \end{align*} $$

In particular, $\mathrm {dens}_f(\beta \beta \beta \ldots )=\frac {\#\beta }{f(\beta )}$ . This is a generalization of the scenario described before by taking $f=|\cdot |$ and $\mathcal {D}=\mathcal {D}_K$ .

Theorem 13. Let $\Sigma $ be a set of symbols and $\mathcal {D}\subset \Sigma ^*\times \Sigma ^*$ be a finite domino set. Let $f:\Sigma ^* \to \mathbb {R}_+$ be a norm function. Assume that $\sup _{\gamma \in \mathcal {T}(\mathcal {D})} \mathrm {dens}_f(\gamma ) \in (0,\infty )$ . Then there exists a periodic linear domino tiling $\gamma \in \mathcal {T}(\mathcal {D})$ , with period $\leq \#\mathcal {D}$ , such that

$$ \begin{align*}\mathrm{ dens}_f(\gamma)=\sup_{\alpha\in\mathcal{T}(\mathcal{D})} \mathrm{dens}_f(\alpha).\end{align*} $$

The following corollary is an immediate consequence of the above theorem by using $\Sigma =K$ and $f=|\cdot |$ .

In order to prove Theorem 13, we need to introduce some terminology. Let $V:= \{w\in \Sigma ^* : \exists w'\ s.t. \ (w,w')\in \mathcal {D} \text { or } (w',w)\in \mathcal {D}\}$ . We then can define a directed graph by $\mathcal {G}=(V,\mathcal {D})$ , with vertex set V and directed edges $\mathcal {D}$ . From now on, in $\mathcal {G}$ , we only consider directed paths (i.e., finite sequences of the form $\gamma =w_1\ldots w_m$ such that $(w_j,w_{j+1}) \in \mathcal {D}$ for any j). We say that the path length of $\gamma $ is m. We also say that $\gamma $ is a closed path when $(w_m,w_1)\in \mathcal {D}$ and open otherwise. An atom is a simple closed path (closed and with no self-intersections).

Proof of Theorem 13.

Since $\mathcal {D}$ is finite, then V is finite and the set of atoms $\mathcal {A}$ is also finite. First, we claim that for any closed path $\beta =w_1\ldots w_m$ in $\mathcal {G}$ , it holds that

$$ \begin{align*} \frac{\#\beta}{f(\beta)} \leq \max_{\alpha\in\mathcal{A}} \frac{\#\alpha}{f(\alpha)}. \end{align*} $$

The inequality is trivial if $\beta $ is an atom. Assume by induction the claim is true if the path length of $\beta $ is $\leq M$ . For $m=M+1$ and $\beta $ not an atom, then $\beta $ must have some self-intersection (i.e., there exist $1\leq i<j\leq M+1$ such that $w_i=w_j$ ). Let $v_1=w_1\ldots w_{i-1}$ , $v_2=w_i\ldots w_{j-1}$ and $v_3=w_j\ldots w_{M+1}$ . Then $v_1,v_2,v_3\in \Sigma ^*$ , $v_1v_3$ and $v_2$ are closed paths in $\mathcal {G}$ with path lengths $\leq M$ (note that we may have $v_1$ or $v_3$ the empty word, but not both at the same time). By monotonicity hypothesis and the induction hypothesis, we deduce

$$ \begin{align*} \frac{\#\beta}{f(\beta)}=\frac{\#(v_1v_2v_3)}{f(v_1v_2v_3)}\leq \max\left\{\frac{\#(v_1v_3)}{f(v_1v_3)},\frac{\#v_2}{f(v_2)}\right\}\leq \max_{\alpha\in\mathcal{A}} \frac{\#\alpha}{f(\alpha)}, \end{align*} $$

as desired. Let now $\varepsilon>0$ be sufficiently small and $\gamma =w_1w_2w_3\ldots \in \mathcal {T}(\mathcal {D})$ be such that $\mathrm { dens}_f(\gamma )+\varepsilon> \sup _{\alpha \in \mathcal {T}(\mathcal {D})} \mathrm {dens}_f(\alpha )>\varepsilon $ . Let $m_n\uparrow \infty $ be a sequence such that

$$ \begin{align*}\mathrm{dens}_f(\gamma)=\lim_{n\rightarrow\infty} \frac{\#(w_1\ldots w_{m_n})}{f(w_1\ldots w_{m_n})}.\end{align*} $$

Since $\#\mathcal {D}$ is finite, there exists a word w which appears infinitely many times in the sequence of $\gamma $ . We may assume such word is $w_1$ since the properties of f imply that

$$ \begin{align*}\lim_{n\rightarrow\infty} \frac{\#(w_1\ldots w_{m_n})}{f(w_1\ldots w_{m_n})} = \lim_{n\rightarrow\infty} \frac{\#(w_i\ldots w_{m_n})}{f(w_i\ldots w_{m_n})} \end{align*} $$

for any i. For each n, we fix a path $\gamma _{(w_{m_n},w)}$ in $\mathcal {G}$ from $w_{m_n}$ to w (such a path exists since w appears infinitely many times in $\gamma $ ). By removing atoms, we may assume that $\gamma _{(w_{m_n},w)}$ is simple, so that $\gamma _{(w_{m_n},w)}$ has path length at most the cardinality of $\mathcal {D}$ . Since $\mathcal {D}$ is finite, we have just finitely many simple paths $\gamma _{(w_{m_n},w)}$ , so $\#\gamma _{(w_{m_n},w)}=O(1)$ . Let $\gamma _{(w_{m_n},w)}^*$ be the path $\gamma _{(w_{m_n},w)}$ without the endpoints $w_{m_n}$ and w, so that $\gamma _{(w_{m_n},w)}=w_{m_n}\gamma _{(w_{m_n},w)}^*w_1$ . Hence, $w_1\ldots w_{m_n}\gamma _{(w_{m_n},w)}^*$ is a closed path. We can now use the properties of f to conclude

$$ \begin{align*} \lim_{n\to\infty} \frac{\#(w_1\ldots w_{m_n}\gamma_{(w_{m_n},w)}^*)}{f(w_1\ldots w_{m_n}\gamma_{(w_{m_n},w)}^*)} = \lim_{n\to\infty} \frac{\#(w_1\ldots w_{m_n})}{f(w_1\ldots w_{m_n})}=\mathrm{dens}_f(\gamma). \end{align*} $$

However, $\frac {\#(w_1\ldots w_{m_n}\gamma _{(w_{m_n},w)}^*)}{f(w_1\ldots w_{m_n}\gamma _{(w_{m_n},w)}^*)}\leq \max _{\alpha \in \mathcal {A}} \frac {\#\alpha }{f(\alpha )}$ . This concludes the proof.

Figure 4 A visualization of the graph $\mathcal {G}$ for $K=\{1,2,\beta \}$ and $3<\beta \leq 4$ via higher-dimensional embedding.

The proof of Theorem 13 does not work for an arbitrary compact $1\in K \subset [1,\infty )$ since it strongly relied on the fact that the graph $\mathcal {G}$ (or the domino set $\mathcal {D}$ ) was finite, so that any closed path could be decomposed into atoms, and hence, in this case, the atoms control the value $\Delta _1(K)$ . Nevertheless, our proof describes an algorithm to find optimal K-admissible sphere packings in the case K is finite.

We now analyse the case when K has 3 elements.

Lemma 15. Let $1<\alpha <\beta $ and $K=\{1,\alpha ,\beta \}$ with $1<\alpha <\beta $ . If $\beta \leq 2$ , the periodic packing $1111\ldots $ has maximal density. If $\beta>2$ , we have the following cases:

5 Proof of Theorem 6

Let $K'=\{\alpha _1,\ldots ,\alpha _N\}$ the set of limit points of K. We can then write K as follows:

$$ \begin{align*} K=\widehat{K}\cup\displaystyle\bigcup_{j=1}^{N} \left\{\alpha_j+\lambda_k^j : k\geq1\right\}, \end{align*} $$

where $\widehat {K}:=\left \{\alpha _1,\ldots ,\alpha _N,\beta _1,\ldots ,\beta _M\right \}$ , with $1\in \widehat {K}$ , and such that

1. (i) $\alpha _i+\alpha _j\not \in \left \{\alpha _r : 1\leq r\leq N\right \};$

2. (ii) $0<\lambda _k^j<\frac {\delta }{2}$ for any $j,k$ , where $\delta :=\displaystyle \min _{1\leq i<j\leq N} \left |\alpha _i-\alpha _j\right |$ ;

3. (iii) For any fixed j, we have $\lambda _k^j\downarrow 0$ when $k\rightarrow \infty $ ;

4. (iv) $\beta _M=\max {K}$ .

We will show that under such conditions, for every packing , one can construct another one $\widetilde P$ with $\mathrm { dens}^+(\widetilde P)\geq \mathrm {dens}^+(P)$ and such that the set of distances of $\widetilde P$ lies in a finite set $\widetilde K \subset K$ . Thus, by Lemma 9 and Corollary 14, there exists a K-admissible periodic sphere packing of maximal density.

For $\gamma \in K$ and $1\leq i,r\leq N$ , we set

$$ \begin{align*} J\left(\gamma,\alpha_i,r\right) :=\left\{s\geq 1 : \gamma+\alpha_i+\lambda_s^r\in K\cup\left(\beta_M,\infty\right)\right\} \ \text{ and } \ M\left(\gamma,\alpha_i,r\right) :=\displaystyle\sup{J\left(\gamma,\alpha_i,r\right)}. \end{align*} $$

Observe that, for fixed $i,j$ , we may have $\alpha _j+\lambda _k^j+\alpha _i=\alpha _w$ for some w. In this case, the index k with such property is unique because of (ii). Therefore, we may write $k(j,i)$ to denote precisely this index k, whenever it exists. Observe also that if $M\left (\gamma ,\alpha _i,r\right )=\infty $ , then $\gamma +\alpha _i$ is an accumulation point and either $\gamma +\alpha _i=\alpha _w$ for some w or $\gamma +\alpha _i\geq \beta _M$ . In particular, if $k\neq k(i,j)$ , then $M\left (\alpha _j+\lambda _k^j,\alpha _i,r\right )=\infty $ if and only if $\alpha _j+\lambda _k^j+\alpha _i\geq \beta _M$ . Our first step is to ask how large $M(\gamma ,\alpha _i,r)$ can be when it is finite.

Step 1. We claim that there is $C>0$ such that for all $1\leq i,j,r\leq N$ and $k\geq 1$ , with $k\neq k(i,j)$ , we have $M\left (\alpha _j+\lambda _k^j,\alpha _i,r\right )\leq C$ whenever it is finite. Indeed, assume by contradiction this does not happen. Then, for each $C\in \mathbb {Z}_+$ there is $s=s(C)\geq C$ , $k\geq 1$ ( $k\neq k(i,j)$ ) and $1\leq i,j,r\leq N$ such that $\alpha _j+\lambda ^j_k+\alpha _i+\lambda ^r_s \in K\cup (\beta _M,\infty )$ but $M\left (\alpha _j+\lambda _k^j,\alpha _i,r\right )<\infty $ . Thus, as $s\to \infty $ , some triple $(i,j,r)$ is repeated infinitely often, and we choose such triple. If $k=k(C)=O(1)$ , then for some $k_0\neq k(i,j)$ , we have that $\alpha _j+\lambda ^j_{k_0}+\alpha _i \in K\cup (\beta _M,\infty )$ is an accumulation point and $M\left (\alpha _j+\lambda _{k_0}^j,\alpha _i,r\right )<\infty $ . Since $k_0\neq k(i,j)$ , we have that $\alpha _j+\lambda ^j_{k_0}+\alpha _i \geq \beta _M$ , but this is absurd since we would have that $M\left (\alpha _j+\lambda _{k_0}^j,\alpha _i,r\right )=\infty $ , which is not the case. Thus, $k=k(C)$ is unbounded, and so $\alpha _j+\alpha _i \in K\cup (\beta _M,\infty )$ is an accumulation point. Condition (i) implies that $\alpha _j+\alpha _i \geq \beta _M$ , and again we reach a contradiction since this implies that $M\left (\alpha _j+\lambda _{k}^j,\alpha _i,r\right )=\infty $ for all k.

Step 2. Let $C>0$ be the constant from the previous step. Enlarge C if necessary so that we also have

$$ \begin{align*}C\geq M(\gamma,\alpha_i,r) \end{align*} $$

for all $\gamma \in \widehat K$ and $1\leq i,r\leq N$ such that $M(\gamma ,\alpha _i,r)<\infty $ . Define the set $\widetilde {K}$ by

$$ \begin{align*} \widetilde{K}:=\widehat{K}\cup\displaystyle\bigcup_{j=1}^{N} \left\{\alpha_j+\lambda_k^j; \text{ } 1\leq k\leq C\right\} \subset K. \end{align*} $$

Let be a packing where $d_i\in K$ . By an inductive argument, we will replace $d_i$ by $\widetilde {d_i}\leq d_i$ , with $\widetilde {d_i}\in \widetilde {K}$ , in such a way that $\widetilde P=\widetilde d_1 \widetilde d_2 \ldots \in \mathcal {P}_+(\widetilde K) \subset \mathcal {P}_+(K)$ . This clearly implies that $\mathrm {dens}^+(\widetilde P)\geq \mathrm {dens}^+(P)$ .

We start with the base case. If $d_1\in \widetilde {K}$ , there is nothing to prove. Assume we have $d_1\not \in \widetilde {K}$ ; hence, $d_1\not \in \widehat {K}$ . Therefore, we have $d_1=\alpha _i+\lambda _s^i$ for some $1\leq i\leq N$ and $s>C$ . We claim that $M(d_2,\alpha _i,i)=\infty $ . Indeed, if it were finite, then $M(d_2,\alpha _i,i)\leq C$ , and hence, $s\leq C$ , a contradiction. Therefore, $M(d_2,\alpha _i,i)=\infty $ ; hence, $d_2+\alpha _i+\lambda _t^i\in K\cup \left (\beta _M,\infty \right )$ for infinitely many indices t. Therefore, we must have $d_2+\alpha _i\in K\cup \left (\beta _M,\infty \right )$ . We now claim that we can replace $d_1=\alpha _i+\lambda _s^i$ by $\widetilde {d_1}:=\alpha _i$ . In other words, we must verify that $\widetilde {d_1}$ is compatible with $d_2,\ldots ,d_{1+\left \lceil \beta _M\right \rceil }$ . This is the reason why we considered the more general sets $J(\gamma ,\alpha _i,r)$ with $r\neq i$ . Since $d_2+\alpha _i\in K\cup \left (\beta _M,\infty \right )$ , it follows that $\widetilde {d_1}$ is compatible with $d_2$ . If $\alpha _i+d_2\geq \beta _M$ , then $\alpha _i+d_2+\ldots +d_w\geq \beta _M$ for any $w\geq 3$ ; hence, $\widetilde {d_1}$ satisfies all the required compatibility conditions. Assume we have $\alpha _i+d_2<\beta _M$ . Since $\alpha _i+d_2+\lambda _t^i\in K\cup \left (\beta _M,\infty \right )$ for infinitely many t, it must be the case in which $\alpha _i+d_2=\alpha _r$ for some r. We claim that $M(d_3,\alpha _r,i)=\infty $ . Indeed, if it were $<\infty $ , then $M(d_3,\alpha _r,i)\leq C$ , and, hence $s\leq C$ , a contradiction. Therefore, we have $\alpha _r+d_3+\lambda _t^i\in K\cup \left (\beta _M,\infty \right )$ for infinitely many t, and hence, $\alpha _i+d_2+d_3=\alpha _r+d_3\in K\cup \left (\beta _M,\infty \right )$ which proves $\widetilde {d_1}$ is also compatible with $d_3$ . If $\alpha _r+d_3\geq \beta _M$ , then we are done. Otherwise, we apply the same procedure to show that $\widetilde {d_1}$ is compatible with $d_4$ , and so on. By repeating this process at most $\left \lceil \beta _M\right \rceil $ times, we conclude that, indeed, it is possible to replace $d_1$ by $\widetilde {d_1}=\alpha _i$ .

Assume that we have replaced $d_1,\ldots ,d_n$ by $\widetilde {d_1},\ldots ,\widetilde {d_n}\in \widetilde {K}$ . If $d_{n+1}\in \widetilde {K}$ , then we are done. If not, we repeat the same procedure as for $d_1$ , but now for both the right- and the left-hand side of $d_{n+1}$ . This completes the induction argument. Observe that $d_n\neq \widetilde {d_n}$ precisely when $d_n=\alpha _i+\lambda _k^i$ , for $k>C$ , and in this case, we took $\widetilde {d_n}=\alpha _i<d_n$ . This process does not reduce the density of the packing.

6 Constructions via modular forms

In this section, we present some constructions via modular forms that generalize the ones in [Reference Cohn and Gonçalves9, Reference Cohn, Kumar, Miller, Radchenko and Viazovska10, Reference Viazovska27] to all dimensions divisible by $4$ .

Throughout the rest of the paper, we will always use z for a variable in the upper-half plane $\mathbb {H}=\{x+it: x\in \mathbb {R}, t>0\}$ . We will use the convention $q=e^{2\pi i z}$ and $r=e^{\pi i z}$ for $z\in \mathbb {H}$ . We will be handling holomorphic modular forms $f:\mathbb {H}\to \mathbb {C}$ over the principal congruence subgroups

$$ \begin{align*}\Gamma(N) = \{\gamma\in \mathrm{SL}_2(\mathbb{Z}) : \gamma \equiv \text{Id} \ (\text{mod } N) \} \end{align*} $$

(note $\Gamma (1)=\mathrm {SL}_2(\mathbb {Z})$ ). A holomorphic modular form of weight k for a subgroup $\Gamma <\Gamma (1)$ is a holomorphic function $f: \mathbb {H} \to \mathbb {C}$ invariant by the slash operation

$$ \begin{align*}f|_k\gamma(z) := (cz+d)^{-k}f\bigg(\frac{az+b}{cz+d}\bigg)=f(z), \ \ \text{ for all } z\in \mathbb{H} \text{ and } \gamma=\left(\begin{matrix} a & b\\ c & d\end{matrix}\right)\in \Gamma, \end{align*} $$

and such that

$$ \begin{align*}f|_k\gamma(z) = O(1) \ \ \text{ as } \mathrm{Im}\, z\to\infty. \end{align*} $$

We denote by $M_k(\Gamma )$ for the space of holomorphic modular forms $f:\mathbb {H}\to \mathbb {C}$ of weight k for $\Gamma $ . We simply write $M_k$ for $M_k(\text {SL}_2(\mathbb {Z}))$ . We let

$$ \begin{align*}T = \left( \begin{matrix} 1 & 1\\ 0 & 1\end{matrix}\right) \ \ \text{and} \ \ S = \left( \begin{matrix} 0 & -1\\ 1 & 0\end{matrix}\right). \end{align*} $$

If $T^2 \in \Gamma $ , then any $f\in M_k(\Gamma )$ has a Fourier r-series of the form

$$ \begin{align*}f(z) = \sum_{n\geq 0} a_n r^n. \end{align*} $$

The Eisenstein series

$$ \begin{align*} E_4(z) = 1+240 \sum_{n\geq 1} \sigma_{3}(n)q^{n} \ \ \text{ and } \ \ E_6(z) =1- 504 \sum_{n\geq 1} \sigma_{5}(n)q^{n}, \end{align*} $$

are classical examples of modular forms in $M_4$ and $M_6$ , respectively. It is easy to see that $M_k$ is trivial if k is odd or if $k<4$ is nonzero, while $M_0$ contains only constants. Indeed, a fundamental result is that

$$ \begin{align*}\bigoplus_{j\in \mathbb{Z}} M_{j} = \mathbb{C}[E_4,E_6]. \end{align*} $$

We will also need Ramanujan’s cusp form of weight $12$

$$ \begin{align*}\Delta(z)= \frac{E_4^3(z)-E_6^2(z)}{1728} = q \prod_{n\geq 1}(1-q^n)^{24} \in M_{12}, \end{align*} $$

which clearly never vanishes for $z\in \mathbb {H}$ . We will also need the Jacobi theta functions defined by

$$ \begin{align*} \Theta_{00}(z) = \sum_{n \in \mathbb{Z}} r^{n^2}, \quad \Theta_{10}(z) = \sum_{n \in \mathbb{Z}} r^{ (n+1/2)^2}\quad \text{and} \quad \Theta_{01}(z) = \sum_{n \in \mathbb{Z}} (-1)^n r^{n^2}. \end{align*} $$

We define their fourth powers by

$$ \begin{align*}U=\Theta_{00}^4, \quad V=\Theta_{10}^4\quad \text{and} \quad W=\Theta_{01}^4. \end{align*} $$

These are modular forms of weight $2$ that satisfy the Jacobi identity

$$ \begin{align*}U=V+W \end{align*} $$

and the transformation laws

$$ \begin{align*} & U|_2T = W, \ \ V|_2T = -V, \ \ W|_2T = U,\\ & U|_2S = -U, \ \ V|_2S = -W, \ \ W|_2S = -V. \end{align*} $$

The functions $U,V,W$ are examples of modular forms of weight $2$ for $\Gamma (2)$ . Another fundamental result is that

$$ \begin{align*}\bigoplus_{j\in \mathbb{Z}} M_{j}(\Gamma(2)) = \mathbb{C}[U,V,W]. \end{align*} $$

Indeed, $M_k(\Gamma (2))$ , for even $k\geq 2$ , coincides with the space of homogeneous polynomials of degree $k/2$ in any two of the $U,V,W$ . Finally, we let

$$ \begin{align*}E_2(z) = 1-24 \sum_{n\geq 1} \sigma_{1}(n)q^{n} \end{align*} $$

be the quasimodular Eisenstein series of weight $2$ . It satisfies the transformation rules

$$ \begin{align*}E_2(z+1)=E_2(z) \text{ and } E_2(-1/z)z^{-2} = E_2(z) + \frac{6}{\pi i z}. \end{align*} $$

We define the space of holomorphic quasimodular forms of weight k and depth p over a subgroup $\Gamma $ by

(2) $$ \begin{align} M_k^{\leq p}(\Gamma) = \bigoplus_{j=0}^p E_2^jM_{k-2j}(\Gamma). \end{align} $$

We again omit $\Gamma $ when $\Gamma =\mathrm {SL}_2(\mathbb {Z})$ . For more details about all these modular forms, we recommend [Reference Zagier28].

The following proposition is key to build admissible functions for the linear programming bounds we have developed.

Proposition 16. Let $\Lambda \subset \mathbb {R}$ be finite, with $0\in \Lambda $ and such that $-1/\lambda \in \Lambda $ whenever $\lambda \in \Lambda $ and $\lambda \neq 0$ . Let

$$ \begin{align*}p(s) = \sum_{\lambda\in \Lambda} a_\lambda e^{\pi i s \lambda} \end{align*} $$

be a trigonometric polynomial. Let $\varepsilon \in \{-1,1\}$ , $d\geq 1$ be an integer and $f:\mathbb {H}\to \mathbb {C}$ be analytic. Suppose that

1. (a) We have

   $$ \begin{align*}\int_0^1|f(it)|t^{-d/2} \mathrm{d} t <\infty; \end{align*} $$

2. (b) There is $\delta>0$ such that for any $c>0$ , we have

   $$ \begin{align*}f(z)=O_c(e^{\pi \delta \Im z}) \end{align*} $$
   if $\Im z> c > |\Re z| $ ;

3. (c) For all $\lambda \in \Lambda \setminus \{0\}$ and $z\in \mathbb {H}$ , we have

   $$ \begin{align*}a_\lambda f(z-\lambda) = -\varepsilon a_{-1/\lambda} (z/i)^{d/2-2}f(-1/z+1/\lambda); \end{align*} $$

4. (d) For all $z\in \mathbb {H}$ , we have

   $$ \begin{align*}\sum_{\lambda \in \Lambda} a_\lambda f(z-\lambda) = \varepsilon a_0 (z/i)^{d/2-2} f(-1/z). \end{align*} $$

Then the function

$$ \begin{align*}h(s) = p(s)\int_{0}^{\infty i} f(z)e^{\pi i z s} \mathrm{d} z \end{align*} $$

defines an analytic function for $\Re s>0$ that extends to an continuous function in $\Re s\geq 0$ , and satisfies the identity

(3) $$ \begin{align} & h(s)\notag \\ & = \sum_{\substack{\lambda\in \Lambda \\ \lambda>0}} \int_\lambda^i a_\lambda f(z-\lambda)(e^{\pi i z s } +\varepsilon (i/z)^{d/2}e^{\pi i (-1/z) s})\mathrm{d} z + a_0\int_0^i f(z)(e^{\pi i z s } + \varepsilon(i/z)^{d/2}e^{\pi i (-1/z) s})\mathrm{d} z \end{align} $$

if $\Re s \geq 0$ . In particular, the function $ F(x)=h(|x|^2) $ belongs to $L^1(\mathbb {R}^d) \cap C^\infty (\mathbb {R}^d)$ and $\widehat F(x) = \varepsilon F(x)$ .

Proof. Conditions (a) and (b) guarantee that $h(s)$ is analytic in the region $\Re s>\delta $ . The analyticity of $h(s)$ for $0<\Re s <2\delta $ and continuity of $h(s)$ for $\Re s=0$ follows straightforwardly by identity (3) and condition (b). The fact that $F(x)$ defines a radial $L^1$ -function and $\widehat F(x) = \varepsilon F(x)$ follows by identity (3), condition (a) and the fact that $x\in \mathbb {R}^d \to (e^{\pi i z |x|^2 } +\varepsilon (i/z)^{d/2}e^{\pi i (-1/z) |x|^2})$ is a eigenfunction of the Fourier transform in $\mathbb {R}^d$ with eigenvalue $\varepsilon $ . It remains to prove identity (3). By analytic continuation (and condition (b)), it is enough to prove it for $\Re s> \delta $ . We then have

$$ \begin{align*} h(s) & = \sum_{\lambda \in \Lambda} a_\lambda \int_{\lambda}^{\lambda+\infty i} f(z-\lambda)e^{\pi i z s}\mathrm{d} z \\ & = \sum_{\substack{\lambda\in \Lambda \\ \lambda \neq 0}} a_\lambda \int_{\lambda}^{i} f(z-\lambda)e^{\pi i z s}\mathrm{d} z + \int_{i}^{\infty i} \big(\sum_{\lambda \in \Lambda} a_\lambda f(z-\lambda)\big) e^{\pi i z s} \mathrm{d} z + a_0\int_0^i f(z) e^{\pi i z s} \mathrm{d} z, \end{align*} $$

where in the second line above, we applied condition (b) to split the integral from $\lambda $ to i and i to $\infty i$ . We obtain

$$ \begin{align*} & h(s) \\ & = \sum_{\substack{\lambda\in \Lambda \\ \lambda> 0}} \bigg(a_\lambda \int_{\lambda}^{i} f(z-\lambda)e^{\pi i z s}\mathrm{d} z + a_{-1/\lambda} \int_{-1/\lambda}^{i} f(z+1/\lambda)e^{\pi i z s}\mathrm{d} z\bigg) \\ & \quad + a_0\int_{i}^{\infty i} \varepsilon (z/i)^{d/2-2} f(-1/z) e^{\pi i z s} \mathrm{d} z + a_0\int_0^i f(z) e^{\pi i z s} \mathrm{d} z \\ & = \sum_{\substack{\lambda\in \Lambda \\ \lambda > 0}} \int_{\lambda}^{i} \big(a_\lambda f(z-\lambda)e^{\pi i z s}\mathrm{d} z + a_{-1/\lambda} f(-1/z+1/\lambda)e^{\pi i (-1/z) s}z^{-2}\big)\mathrm{d} z \\ & \quad + a_0\int_{0}^{i}\big(\varepsilon (i/z)^{d/2} e^{\pi i (-1/z) s} + e^{\pi i z s}\big)f(z) \mathrm{d} z \\ & = \sum_{\substack{\lambda\in \Lambda \\ \lambda > 0}} \int_{\lambda}^{i} a_\lambda f(z-\lambda)\big(e^{\pi i z s} + \varepsilon(i/z)^{d/2}e^{\pi i (-1/z) s}\big)\mathrm{d} z + a_0\int_{0}^{i}\big(\varepsilon (i/z)^{d/2} e^{\pi i (-1/z) s} + e^{\pi i z s}\big)f(z) \mathrm{d} z, \end{align*} $$

where in the first equality, we have applied condition (d) and that $\Lambda $ is closed by $\lambda \to -1/\lambda $ , in the second equality, we used the change of variables $z\to -1/z$ , and in the last, we have applied condition (c).

Remark 2. It is straightforward to show that if we strengthen condition (a) to

$$ \begin{align*}\int_0^1|f(it)|t^{-k} \mathrm{d} t <\infty; \end{align*} $$

for all $k>0$ , then $\partial _{x}^\alpha F \in L^1(\mathbb {R}^d)$ for all multi-indexes $\alpha \in \mathbb {Z}_+^d$ , and since $\widehat F=\varepsilon F$ , we conclude that F is of Schwartz class.

This integral transform was used by Viazovska et al. in [Reference Cohn, Kumar, Miller, Radchenko and Viazovska10, Reference Viazovska27] for $p(s)=\sin ^2(\pi s/2)$ to build the magic functions for the packing problem, and also by Radchenko and Viazovska in [Reference Radchenko and Viazovska23], for $p(s)=\sin (\pi s)$ , to construct Fourier interpolation formulae with $\sqrt {n}$ -nodes. For our purposes, we wish to investigate the case when $p(s)=\sin ^2(\pi s/2)$ . The following lemmas generalize the constructions in [Reference Cohn, Kumar, Miller, Radchenko and Viazovska10, Reference Viazovska27] to all dimensions multiple of $4$ .

Remark 3. To make computations simpler we will, from now on, adopt the nonstandard notation that

$$ \begin{align*}f_T(z) =f|_kT(z)=f(z+1) \text{ and } f_S(z)=f|_kS(z)=f(-1/z)z^{-k}, \end{align*} $$

whenever it can be inferred from the context that f has weight k; that is, $f\in M_k^{\leq p}(\Gamma )$ , for some $\Gamma $ , k and p.

Lemma 17 ( $\Gamma (2)$ -construction).

Let $d\geq 4$ be an integer divisible by $4$ . Let $l\geq d/12$ be an even integer and set

$$ \begin{align*}k=2-d/2+6l \text{ and } \varepsilon=(-1)^{d/4+1}. \end{align*} $$

Consider the following linear operatorFootnote ⁴

$$ \begin{align*} L: \varphi \in M_k(\Gamma(2)) & \mapsto (\varphi_T+\varphi_S - \varphi,\, [r^0] \varphi_S,\, [r]\varphi_S, \ldots , \, [r^l]\varphi_S). \end{align*} $$

Then any function in the vector space

$$ \begin{align*}\mathcal{F}_{d,l}:=\frac{1}{\Delta^{l/2}}\ker(L) \end{align*} $$

satisfies all conditions in Proposition 16 for $\varepsilon $ , d and $p(s)=\sin ^2(\tfrac {\pi }{2} s)$ . In particular, for any $f\in \mathcal {F}_{d,l}$ , we have an associated radial Schwartz function $F_f:\mathbb {R}^d\to \mathbb {C}$ such that $\widehat {F_f}=\varepsilon F_f$ and

$$ \begin{align*}F_f(x) = \sin^2(\tfrac{\pi}{2} |x|^2)\int_0^{\infty i} f(z)e^{\pi i z |x|^2}\frac{\mathrm{d} z}{i} \end{align*} $$

if $|x|^2>l$ .

Proof. We will show that all conditions in Proposition 16 are satisfied for $p(s)=\sin ^2(\tfrac {\pi }{2}s)$ . First, observe that if $f\in \mathcal {F}_{d,l}$ , then condition (a) in Proposition 16 is equivalent to

$$ \begin{align*}\int_{i}^{\infty i} |f|_kS(z)||z|^{6l} |\mathrm{d} z| <\infty. \end{align*} $$

This is true because $f=\varphi /\Delta ^{l/2}$ and

$$ \begin{align*}z^{6l}f|_kS =\frac{\varphi_S(z)}{\Delta^{l/2}(z) }, \end{align*} $$

and so we conclude the Fourier expansion of $z^{6l}f|_kS$ starts at r. Condition (b) in Proposition 16 is trivial since f is $2$ -periodic and has Fourier expansion starting at $r^{-l}$ . A simple computation shows that condition (d) implies (c) under the $2$ -periodicity of f. Using the modular properties of $\Delta (z)$ , we see that condition (d) in Proposition 16 is equivalent to

$$ \begin{align*}\varphi = \varphi_S+\varphi_T, \end{align*} $$

which holds true.

Remark 4. It is worth pointing out that by the fact that (see [Reference Eichler and Zagier13, Remark after Theorem 8.4] and [Reference Cohn, Kumar, Miller, Radchenko and Viazovska11, Equation (2.13)])

$$ \begin{align*}M_k(\Gamma(2)) = M_k\oplus UM_{k-2}\oplus VM_{k-2}\oplus U^2M_{k-4}\oplus V^2M_{k-4}\oplus UVWM_{k-6}, \end{align*} $$

routine computations show that the kernel of $\varphi \mapsto \varphi _T+\varphi _S - \varphi $ in $M_k(\Gamma (2))$ coincides with

$$ \begin{align*} (U-V)M_{k-2} \oplus (U^2 - V^2)M_{k-4}, \end{align*} $$

which has dimension $ \lceil k/6 \rceil $ for $k\geq 6$ , but is trivial otherwise. One can also show that

$$ \begin{align*}\dim \mathcal{F}_{d,l}=\bigg\lceil \frac{k}{6} \bigg\rceil-\frac{l}{2}=\frac{l}{2} -\bigg \lfloor \frac{d-4}{12} \bigg\rfloor. \end{align*} $$

Lemma 18 ( $\Gamma (1)$ -construction).

Let $d\geq 4$ be an integer divisible by $4$ . Let $l\geq d/12$ be an even integer and set

$$ \begin{align*}k=2-d/2+6l \quad \text{ and } \quad \varepsilon=(-1)^{d/4}. \end{align*} $$

Consider the following linear operator:

$$ \begin{align*} L: \psi \in M_{k+2}^{\leq 2} & \mapsto ([q^0] \psi,\, [q^{1}]\psi, \ldots , \, [q^{l/2}]\psi). \end{align*} $$

Then any function in the vector space

$$ \begin{align*}\mathcal{G}_{d,l}:=\frac1{\Delta^{l/2}}(z^2\ker(L)_S), \end{align*} $$

where $z^2\ker (L)_S:=\{z\mapsto z^2 \psi |_{k+2}S(z): \psi \in \ker (L)\}$ , satisfies all conditions in Proposition 16 for $\varepsilon $ , d and $p(s)=\sin ^2(\tfrac {\pi }{2} s)$ . In particular, for any $g\in \mathcal {G}_{d,l}$ , we have an associated radial Schwartz function $G_g:\mathbb {R}^d\to \mathbb {C}$ such that $\widehat {G_g}=\varepsilon G_g$ and

$$ \begin{align*}G_g(x) =\sin^2(\tfrac{\pi}{2} |x|^2)\int_0^{\infty i} g(z)e^{\pi i z |x|^2}\frac{\mathrm{d} z}{i} \end{align*} $$

if $|x|^2>l$ .

Proof. We will show that all conditions in Proposition 16 are satisfied for $p(s)=\sin ^2(\tfrac {\pi }{2} s)$ . Conditions (a) and (b) follow in a similar manner as in the proof of Lemma 17. Setting $g=\psi |_kS/\Delta ^{l/2}$ , condition $(c)$ for g is then equivalent to

$$ \begin{align*}\psi|_kST^{-1}=\varepsilon(-1)^{d/4}\psi|_kSTS, \end{align*} $$

which is true since $\psi $ is $1$ -periodic, $\varepsilon (-1)^{d/4}=1$ and $ST^{-1}(STS)^{-1}=T$ . Condition $(d)$ for g is equivalent to

$$ \begin{align*}\widetilde \psi(z)-\tfrac12(\widetilde \psi(z+1)+\widetilde \psi(z-1)) = \psi(z), \end{align*} $$

where $\widetilde \psi (z) =\psi |_kS(z)$ . However, simple computations show that this equation holds true for any function $\psi \in M_{k+2}^{\leq 2}$ using the characterization (2) and the functional equation of $E_2(z)$ .

Remark 5. It is not hard to show any function $g\in \mathcal {G}_{d,l}$ has an q-expansion of the form

$$ \begin{align*}g=\sum_{n= -l/2}^\infty a_n q^{n}, \end{align*} $$

where $a_n$ is a quadratic polynomial in z for $n\geq 1$ , but affine for $n=-l,\ldots ,0$ .

The following lemma is reminiscent of the numerical method employed in [Reference Cohn, Kumar, Miller, Radchenko and Viazovska10].

Lemma 19 (Effective tail bounds).

Let $P(X,Y,Z)$ and $Q(X,Y,Z)$ be homogeneous polynomials of degree $k/2$ and $k+2$ , respectively, where k is even. Let $|P|$ and $|Q|$ denote the homogeneous polynomials derived from P and Q, where each coefficient is replaced by its absolute value. Assume that Q has no power of X larger than $2$ . Define the following holomorphic modular forms:

$$ \begin{align*}\varphi= P(U,V,W) \ \ \text{ and } \ \ \psi= Q(E_2,E_4,E_6). \end{align*} $$

Let $\varphi _M$ and $\psi _M$ denote the tail of their respective r-series and q-series from the $(M+1)$ -term onward. Let

$$ \begin{align*}\varphi_S= P(U_S,V_S,W_S) \end{align*} $$

and $(\varphi _S)_M$ be the tail of the r-series of the above function from the $r^{M+1}$ -term Furthermore, letting $w=-\pi i z$ and

$$ \begin{align*}w^2 \psi_S = w^2 Q((E_2)_S,E_4,E_6), \end{align*} $$

denote by $(w^2\psi _S)_M$ the tail of the q-series above from the $q^{M+1}$ -term. Finally, let

$$ \begin{align*}R_M(p,j)=\sum_{n>M}{(n+1)^p}e^{-\pi{(n-j)}} \ \ \text{ and } \ \ S_M(p,j)=\sum_{n>M}{(n+1)^p}e^{-2\pi{(n-j)}}. \end{align*} $$

Then for $j\leq M+1$ , $t\geq 1$ , $r=e^{-\pi t}$ and $q=r^2$ , we have

1. (1) $|\varphi _M(it)| \leq |P|(8,8,8) R_M(\tfrac {3k-2}{2},j)r^j$ ;

2. (2) $|\psi _M(it)| \leq |Q|(24,240,504) S_M(\tfrac {5k+10}{4},j)q^j$ ;

3. (3) $|(\varphi _S)_{M}(it)| \leq |P|(8,8,8)R_{M}(\tfrac {3k-2}{2},j)r^{j}$ ;

4. (4) $|(w^2 \psi _S)_M(it)| \leq 13|Q|(24,240,504)|S_M(\tfrac {5k+10}{4},j)q^j$ .

Proof. First we prove (1). Observe that by Jacobi’s four-square theorem, the coefficient of $r^n$ in the r-series of each of the functions $U,V$ and W is bounded by $8(n+1)^2$ . Also note that whenever we multiply m power series $\sum _{n\geq 0} (n+1)^{a_j} r^n$ for $j=1,\ldots ,m$ , the coefficient of $r^n$ in the product is bounded by $(n+1)^{a_1+\ldots +a_m+m-1}$ . We deduce that the coefficient of $r^n$ in the r-series of $\varphi $ is bounded by $|P|(8,8,8) (n+1)^{(3k-2)/2}$ . Since $t\geq 1$ , we have $0\leq r\leq e^{-\pi }$ . This easily implies item (1). The same argument shows item (3). Essentially the same argument shows item (2) by realizing that the coefficient of $q^n$ in the q-series of each of the functions $E_2,E_4$ and $E_6$ are bounded by $24(n+1)^2$ , $240(n+1)^4$ and $504(n+1)^6$ , respectively. We deduce that the coefficient of $q^n$ in the q-series of $\psi $ is bounded by $ |Q|(24,240,504) (n+1)^{a+b+c+k+1}$ , where $a+b+c$ is maximal among nonnegative integers $a,b$ and c such that $2a+4b+6c=k+2$ and $0\leq a\leq 2$ . Greedy choice shows that the sum $a+b+c$ is maximized (or bounded by) when $(a,b,c)=(2,(k-2)/4,0)$ , and thus, the coefficient of $q^n$ is bounded by $|Q|(24,240,504) (n+1)^{(5k+10)/4}$ . This proves item (2). Finally, item (4) is more nuanced since we have the presence of w. Let

$$ \begin{align*}Q(X,Y,Z) = Q_0(Y,Z) + XQ_1(Y,Z)+X^2 Q_2(Y,Z), \end{align*} $$

so that $|Q(24,240,504)|=|Q_0|(240,504)+24|Q_1|(240,504)+24^2|Q_2|(240,504)$ and

$$ \begin{align*} w^2 \psi_S & = w^2Q_0(E_4,E_6) + ({w^2}E_2-{6w})Q_1(E_4,E_6) + ({w}E_2-6)^2Q_2(E_4,E_6) \\ & = Qw^2 + (-12Q_2E_2 - 6Q_1)w + 36Q_2. \end{align*} $$

For $z=it$ and $t\geq 1$ , we have $\pi \leq w=\pi t \leq \tfrac 1{7r}$ , and we obtain

$$ \begin{align*} {|(w^2 \psi_S)_M(it)|} & \leq (\tfrac1{7r})^2{|Q|(24,240,504) }S_M(\tfrac{5k+10}{4},j+1)q^{j+1} \\ & \quad + \tfrac1{7r} (288|Q_2|(240,504)+6|Q_1|(240,504))S_M(\tfrac{5k}{4},j+1/2)q^{j+1/2}\\ & \quad +36|Q_2|(240,504)S_M(\tfrac{5k-10}{4},j)q^j \\ & \leq (\tfrac{e^{2\pi}}{49}{|Q|(24,240,504) } + \tfrac{e^{\pi}}{7} (288|Q_2|(240,504)+6|Q_1|(240,504))\\ & \quad +36|Q_2|(240,504))S_M(\tfrac{5k+10}{4},j)q^j \\ & < (11|Q_0|(240,504) + 283|Q_1|(240,504) + 7283|Q_2|(240,504))S_M(\tfrac{5k+10}{4},j)q^j\\ & <13|Q|(24,240,504)S_M(\tfrac{5k+10}{4},j)q^j \end{align*} $$

This proves item (4).

7 Proof of Theorem 1

We assume Theorem 4. Let $d=48$ and $K=\frac {1}{\sqrt {6}}([\sqrt {6},\sqrt {8}]\cup \{\sqrt {10}\})$ . Let $\Lambda \subset \mathbb {R}^{48}$ be an even unimodular extremal lattice. In particular, $\Lambda $ is self-dual and $\ell (\Lambda )=\{\sqrt {6},\sqrt {8},\ldots \}$ . It is trivial to see $\Lambda $ is K-admissible and that

$$ \begin{align*}\mathrm{dens}(\Lambda)= \mathrm{vol} \left( B_{48}\right) \left( \frac{\sqrt{6}}{2}\right)^{48}=\frac{(3\pi/2)^{24}}{24!}. \end{align*} $$

Consider $F(x)=H(\sqrt {6}x)$ , with H as in Theorem 4 for $d=48$ . Poisson summation over $\Lambda $ shows that $H(0)=\widehat H(0)>0$ . The properties of H imply that F satisfies all conditions of Theorem 3, and hence,

$$ \begin{align*}\mathrm{dens}(\Lambda) \leq \Delta_d(K) \leq \mathrm{vol}(\tfrac12B_{48} ) \frac{F(0)}{\widehat F(0)}= \mathrm{vol} \left( B_{48}\right) \left( \frac{\sqrt{6}}{2}\right)^{48}. \end{align*} $$

This shows that equality above is attained and $\Lambda $ is optimal; that is, $\Delta _d(K) = \mathrm {dens}(\Lambda )$ .

Now we prove uniqueness among all periodic packings. We follow the same strategy as in [Reference Cohn and Elkies8, Section 8]. Let $P=L+Y+\frac {1}{2}B_d$ be an optimal admissible periodic packing for some lattice L and a set of translations $Y=\{v_1,\ldots ,v_M\}$ . By Poisson Summation,

$$ \begin{align*} \sum_{j,l=1}^{M} \sum_{x\in L} F(x+v_j-v_l)=\frac{1}{|L|}\sum_{y\in L^*} \widehat{F}(y)\left|\sum_{j=1}^{M} e^{2\pi i yv_j}\right|^2, \end{align*} $$

from which we derive

$$ \begin{align*} M F(0)\geq\frac{1}{|L|}\widehat{F}(0) M^2. \end{align*} $$

Since P is optimal, we have equality above, from which we derive that $|\sqrt {6}L|=M$ and that $\{x+v_j-v_l : x\in L, \, 1\leq j,l\leq M\}$ is contained in the set of zeros of F. By Theorem 4, we deduce that $\{\sqrt {6}|x+v_j-v_l| : x\in L, \, 1\leq j,l\leq M\}\subset \{0,\sqrt {6},\sqrt {8},\sqrt {10},\ldots \}$ . By [Reference Cohn and Elkies8, Lemma 8.2], we conclude that the subgroup G of $\mathbb {R}^{48}$ generated by the set $\sqrt {6}(L+Y)$ is an even integral lattice with minimal norm $\geq \sqrt {6}$ . It also follows that the volume $|G|=\sqrt {N}$ , for some integer N, and hence, G has at most one point per unit of volume in $\mathbb {R}^{48}$ . However, since $|\sqrt {6}L|=M$ , the packing $\sqrt {6}(L+Y)+\frac {\sqrt {6}}{2}B_d$ has one sphere per unit of volume. Therefore, G has exactly one point per unit volume, which implies that $N=1$ , G is unimodular and $G=\sqrt {6}(L+Y)$ . Therefore, G must be an extremal lattice. This finishes the proof of Theorem 1.

8 Proof of Theorem 4

Let $8\leq d \leq 1200$ be a multiple of $8$ , $a=a_d$ , $l= l_d$ , $c=c_d$ and $ K=K_d=\frac {1}{\sqrt {a}}\{\sqrt {a},\sqrt {a+2},\ldots ,\sqrt {l}\} $ . In what follows, we set $k=2-d/2+6l$ (which is congruent to $2$ modulo $4$ ), $\varepsilon =-1$ ,

$$ \begin{align*}\mathfrak{b}=\frac{l}{2} - \bigg\lfloor \frac{d-4}{12} \bigg\rfloor, \end{align*} $$

$w=-\pi i z$ , $r=e^{\pi i z}$ and $q=r^2$ . We will abuse notation and write $u(z)=O(r^k)$ if for some $C>0$ , we have $|u(z)|\leq C |z|^C e^{-\pi k \Im z }$ for $\Im z>C$ .

The assertions below were done with rational arithmetic via PARI/GP [Reference Batut, Belabas, Benardi, Cohen and Olivier3] computer algebra system. Below, we will make certain claims, and we will indicate precisely how to prove them. This proof is computer assisted; hence, the necessary ancillary files can be found with the arXiv submission of this paper (arXiv:2308.03925).

Step 1. We apply Lemma 17 and compute a basis for the vector space $\mathcal {F}_{d,l}$ collected as a row vector of functions

$$ \begin{align*}[f_1,\ldots ,f_{\mathfrak{b}}] = \Delta^{-l/2} [\varphi_1,\varphi_2,\ldots ,\varphi_{\mathfrak{b}}], \end{align*} $$

where $\dim \mathcal {F}_{d,l} = \mathfrak {b}$ and

$$ \begin{align*} \Phi &= [\varphi_1,\varphi_2,\ldots ,\varphi_{\mathfrak{b}}] = W^{l+1}(\mathcal{M}_\varphi \mathbf{\Theta}^\top)^\top, \\ \mathcal{M}_\varphi & \in \mathbb{Q}^{\mathfrak{b} \times (k/2-l )}, \\ \mathbf{\Theta} & = [W^{j-1} V^{k/2-l-j}]_{j=1,\ldots ,k/2-l}. \end{align*} $$

We use the symbol $^\top $ for transpose. $\Theta $ is a row vector of basis functions for $M_{k-2l-2}(\Gamma (2))$ . It is easy to show that any function in $\Delta ^{l/2}\mathcal {F}_{d,l}$ must be divisible by $W^{l+1}$ , and this is the reason why we have isolated it in $\Phi $ . To make sure $\Phi $ is uniquely defined (and so $\mathcal {M}_\varphi $ ), we normalize $\varphi _j$ so that

$$ \begin{align*}{\varphi_j} = r^{2(j-1)} + O(r^{2\mathfrak{b}}). \end{align*} $$

Step 2. We apply Lemma 18 and then proceed to find a basis

$$ \begin{align*}[g_1,\ldots ,g_{\mathfrak{c}}] = \Delta^{-l/2} [w^2(\psi_1)_S,\ldots ,w^2(\psi_{\mathfrak{c}})_S], \end{align*} $$

collected as a row vector, for the subspace of functions $g\in \mathcal {G}_{d,l}$ , with $g=w^2 \psi _S/\Delta ^{l/2}$ , and such that

$$ \begin{align*}[w r^{j}](w^2 \Delta^{-l/2} \psi_S) =0 \end{align*} $$

for $j\in \{-l,\ldots ,-a\}$ . By the choice of l and a, it turns out that $\mathfrak {c}=\mathfrak {b}$ for all cases, we have computed (the proof that these dimensions coincide for all d is lengthy and not worth to include here since we are doing this numerically anyways). Here, we set

$$ \begin{align*} \Psi & = [\psi_1,\psi_2,\ldots ,\psi_{\mathfrak{b}}] = (\mathcal{M}_\psi \mathbf{E}^\top)^\top, \\ \mathcal{M}_\psi & \in \mathbb{Q}^{ \mathfrak{b} \times (k+6)/4},\\ \mathbf{E} & = [E_2^{i}E_4^{j}E_6^n]_{\substack{2i+4j+6n=k+2 \\ j =0,\ldots ,(k+2)/4}}. \end{align*} $$

$\textbf {E}$ is a row vector of size $(k+6)/4$ that contains the basis functions for $M_{k+2}$ (for each $j=0,..,(k+2)/4$ , the tuple $(i,n)$ is given by $i=((k+2)/2-2j) \text { mod } 3$ and $n=(k+2-2i-4j)/6$ ). To make sure that $\Psi $ is uniquely defined, we impose that

$$ \begin{align*}{w^2(\psi_j)_S} = r^{2(j-1)} + O(r^{2\mathfrak{b}}) \end{align*} $$

for $j=1,\ldots ,\mathfrak {b}$ .

Step 3. We now solve a linear system of homogeneous equations and set

$$ \begin{align*} \varphi = \mathcal{V}_\varphi \mathcal{M}_\varphi {\boldsymbol \Theta}^\top\ \text{ and } \ \psi & = \mathcal{V}_\psi \mathcal{M}_\psi \mathbf{E}^\top, \end{align*} $$

where $ \mathcal {V}_\varphi $ and $\mathcal {V}_\psi $ are row vectors of size $\mathfrak {b}$ (the solutions) that enforce the following r-expansion shapes:

$$ \begin{align*} {-w^2\psi _S- \varphi} &= \sum_{n= 0}^{l-a} \alpha_n r^{n} +\sum_{n=l - a+1}^{l} (\alpha_n + \alpha_n'w) r^{n} + O(r^{l+1}),\\ {-w^2\psi_S + \varphi} &= \sum_{n= l - a+1}^{l} (\beta_n + \beta'_{n}w) r^{n} + O(r^{l+1}), \end{align*} $$

for some $\alpha _n,\alpha _n',\beta _n,\beta _n'\in \mathbb {Q}$ . Recall that by Remark 5, $[w^2 r^j](-w^2\psi _S)=0$ for $j\leq 0$ . Note also that, by construction, we already have that $\alpha _n,\alpha _n',\beta _n,\beta _n'$ vanish for odd n in the range $0\leq n\leq l$ . More precisely, given all the previous constraints, $ \mathcal {V}_\varphi $ and $\mathcal {V}_\psi $ are solutions of the homogeneous equations

$$ \begin{align*}[w^0 r^j]({-w^2\psi_S + \varphi})=0 \end{align*} $$

for $j\in \{0,2,\ldots ,l-a\}$ . It turns out that $\mathcal {V}_\varphi $ and $\mathcal {V}_\psi $ are uniquely defined modulo scaling. We then define the row vectors

$$ \begin{align*}C_\varphi = {n}\mathcal{V}_\varphi \mathcal{M}_\varphi \ \text{ and } \ C_\psi={n}\mathcal{V}_\psi \mathcal{M}_\psi, \end{align*} $$

where we choose $n\in \mathbb {Z}$ so that $C_\varphi $ and $C_\psi $ are vectors of integers where $\gcd (C_\varphi \cup C_\psi )=1$ and the first nonzero coordinate of $C_\varphi $ is positive. In this way, $C_\varphi $ and $C_\psi $ are uniquely defined. For instance, for $d=48$ , we have

$$ \begin{align*} C_\varphi &=2^7\times 3^8 \times [29393, 117572, 307819, 511955, 539410, 362729, 152114, 36480, 3840] \\ C_\psi & = [565675, 7394933, -38880096, 44550063, 41316945, -107522880, 39169185, \\ & \quad \ \ 40077567, -32756064, 5294597, 790075]. \end{align*} $$

These vectors cannot be simplified much further nor have some easy to guess combinatorial formula since, for instance, the $8$ th entry of $C_\psi $ is divisible by the large prime $4453063$ and $7$ th entry of $C_\varphi $ is divisible by the prime $4003$ . Experimentally, large primes are often found in the vectors $C_\varphi $ and $C_\psi $ as dimension grows. A list of all vectors $C_\varphi $ and $C_\psi $ for each dimension $d\leq 1200$ multiple of $8$ can be found on the ancillary files in the arXiv submission of this paper (a file named Cvectors).

Step 4. We use Lemmas 17 and 18 to create a radial Schwartz function $H:\mathbb {R}^d\to \mathbb {R}$ and obtain the integral representations

$$ \begin{align*} H(x)& =\sin^2(\tfrac{\pi}{2} |x|^2)\int_0^{\infty } \frac{-\pi^2 t^2\psi_S(it)-\varphi(it)}{\Delta^{l/2}(it)}e^{- \pi t |x|^2}\mathrm{d} t \\ \widehat H(x)& =\sin^2(\tfrac{\pi}{2} |x|^2)\int_0^{\infty} \frac{-\pi^2 t^2\psi_S(it)+\varphi(it)}{\Delta^{l/2}(it)}e^{ -\pi t |x|^2}\mathrm{d} t, \end{align*} $$

that hold for $|x|^2>l$ . This shows that

$$ \begin{align*}H(x)=\widehat H(x)=0 \text{ for } |x|^2 \in \{l+2,l+4,\ldots \}. \end{align*} $$

However, if we let $g=-\Delta ^{-l/2} w^2 \psi _S$ and $f=\Delta ^{-l/2} \varphi $ , then, by construction, the $r^j$ -coefficient of $g-f$ is a rational number for $j=-l,\ldots ,-a$ . Similarly, by construction, the $r^j$ -coefficient of $g+f$ vanishes for $j=-l,\ldots ,-a$ . A straightforward computation shows also that $H(x)=\widehat H(x)=0$ for $|x|^2\in \{a,a+2,\ldots \}$ , and the integral representation of $\widehat H(x)$ above converges for $|x|^2>a-2$ .

Step 5. From now on, we assume that . We claim that

1. (i) $|\pi ^2 \psi (it)|< -\varphi _S(it)$ for $t\geq 1$ ;

2. (ii) $|\pi ^2 t^2\psi _S(it)| < \varphi (it)$ for $t\geq 1$ .

Notice that for $|x|^2>a-2$ , we have

$$ \begin{align*} \frac{\widehat H(x)}{\sin^2(\tfrac{\pi}{2} |x|^2)} & = \int_1^{\infty } \frac{-\pi^2 t^2\psi_S(it)+\varphi(it)}{\Delta^{l/2}(it)}e^{- \pi t |x|^2}\mathrm{d} t + \int_1^{\infty } \frac{-\pi^2\psi(it)-\varphi_S(it)}{\Delta^{l/2}(it)}e^{- \pi |x|^2/t} \frac{\mathrm{d} t}{t^{d/2}}> 0, \end{align*} $$

by conditions (i) and (ii). This implies that $\widehat H(x) \geq 0$ for $|x|^2>a-2$ and vanishes exactly at $|x|^2 \in \{a,a+2,a+4,\ldots \}$ if $|x|^2>a-2$ . Similarly, for $|x|^2>l$ , we have

$$ \begin{align*} \frac{H(x)}{\sin^2(\tfrac{\pi}{2} |x|^2)} & = \int_1^{\infty } \frac{-\pi^2 t^2\psi_S(it)-\varphi(it)}{\Delta^{l/2}(it)}e^{- \pi t |x|^2}\mathrm{d} t + \int_1^{\infty } \frac{-\pi^2\psi(it)+\varphi_S(it)}{\Delta^{l/2}(it)}e^{- \pi |x|^2/t} \frac{\mathrm{d} t}{t^{d/2}} < 0, \end{align*} $$

and so $H(x) \leq 0$ for $|x|^2>l$ and vanishes exactly at $|x|^2 \in \{l,l+2,l+4,\ldots \}$ if $|x|^2>l-\varepsilon $ , for some small $\varepsilon>0$ .

Step 6. To prove the claims (i) and (ii) in Step 5, we introduce the following notation: For a given $u=\sum _{n\geq 0} a_n(w) r^n$ , we write

$$ \begin{align*}{(u)^{\mathrm{trunc}}} = \sum_{n= 0}^N a_n(w) r^n + r^{l+10} \ \text{ and } \ {(u)_{\mathrm{trunc}}} = \sum_{n= 0}^N a_n(w) r^n - r^{l+10}. \end{align*} $$

Recall from Lemma 19 that $R_N$ and $S_{N/2}$ are the corresponding tail sums. We will choose $N \geq l+10n$ , with $n\geq 1$ , to be the first integer such that the quantity

$$ \begin{align*}\max\bigg\{\mathrm{abs}(C_\varphi)\mathbf{\Theta}^\top|_{r=0}R_N((3k-2)/2,l+10), 13\mathrm{abs}(C_\psi )\mathbf{ E}^\top|_{q=0}S_{N/2}((5k+10)/4,l/2+5)\bigg\} \end{align*} $$

is less than $1$ . Above, $\mathrm {abs}(v)$ , for a vector v, is simply the same vector with each coordinate replaced by its absolute value. By Lemma 19, this guarantees that

$$ \begin{align*} {(\varphi)_{\mathrm{trunc}}} &\leq \varphi \leq {(\varphi)^{\mathrm{trunc}}}, \quad \quad \quad \, {(\varphi_S)_{\mathrm{trunc}}} \leq \varphi_S \leq {(\varphi_S)^{\mathrm{trunc}}},\\ {(\psi)_{\mathrm{trunc}}} &\leq \psi \leq {(\psi)^{\mathrm{trunc}}}, \quad \quad {(w^2\psi_S)_{\mathrm{trunc}}} \leq w^2\psi_S \leq {(w^2\psi_S)^{\mathrm{trunc}}}, \end{align*} $$

for $z=it$ , $w=\pi t$ and $t\geq 1$ . A list of all N’s for each dimension d can be found on the ancillary files in the arXiv version of this paper (a file named Nnumbers). For instance, for $d=48$ , we have $N=130$ .

To prove that condition (i) is satisfied, first we verify thatFootnote ⁵

1. (I) $ C_\varphi \geq 0$ ,

since it directly shows that $\varphi (it)>0$ for $t>0$ , which implies that (since $k/2$ is odd) $\varphi _S(it)<0$ for $t>0$ . Next, we show thatFootnote ⁶

1. (II) ${(\psi )_{\mathrm {trunc}}}$ has only nonnegative coefficents in its q-expansion,

which proves that $\psi (it)>0$ for $t\geq 1$ and that $-\pi ^2 \psi (it)<-\varphi _S(it)$ for $t\geq 1$ . Next, we use Sturm’s method (which can be done via exact rational arithmetic evaluations) on the variable r to show that the polynomial

1. (III) $- \pi _2^2 {(\psi )^{\mathrm {trunc}}} - {(\varphi _S)^{\mathrm {trunc}}}> 0$ for $0 < r< \gamma _2$ ,

where we use $\pi _2 = \lceil \pi 10^{m}\rceil 10^{-m}$ for $m=20,40,\ldots ,100$ and $\gamma _2=\lceil e^{-\pi } 10^{m'}\rceil 10^{-m'}$ for $m'=2,5,8,11$ , where we select $(m,m')$ according to necessity (for large dimensions, more precision is sometimes required). From now on, whenever we apply Sturm’s method in the range $0 < r< \gamma _2$ , we will select $\gamma _2$ as before. This shows that $\pi ^2 \psi (it) <- {\varphi _S}(it)$ for $t\geq 1$ , and proves that condition (i) holds.

Step 7. For condition (ii), we write

$$ \begin{align*} {(w^2 \psi_S)_{\mathrm{trunc}}}+{(\varphi)_{\mathrm{trunc}}} & = [1,w,w^2][P_0(e^{-w}),P_1(e^{-w}),P_2(e^{-w})]^\top\\ -{(w^2 \psi_S)^{\mathrm{trunc}}}+{(\varphi)_{\mathrm{trunc}}} & = [1,w,w^2][Q_0(e^{-w}),Q_1(e^{-w}),Q_2(e^{-w})]^\top, \end{align*} $$

where the $P_i$ ’s and $Q_i$ ’s are polynomials with integer coefficients and degree at most N. Note that $P_2(e^{-w})={(\psi )_{\mathrm {trunc}}} (it)$ and $Q_2(e^{-w})=-{(\psi )_{\mathrm {trunc}}} (it)-e^{-w(l+10)}$ , so $-Q_2$ and $P_2$ have only positive coefficients. We then set $x=e^{-w}$ and

$$ \begin{align*} w_1(x) =\sum_{n=1}^N \frac{(1-x)^n}{n} \text{ and } w_2(x) = \sum_{n=1}^N \frac{(1-x)^n}{n}+\frac{(1-x)^{N+1}}{(N+1)x}, \end{align*} $$

and note that $w_1(x)<w < w_2(x)$ for $0<x<1$ . We then conclude that condition (ii) is implied by the two conditions below:

1. (IV) $[1,w_{j}(x),w_{1}(x)^2][P_0(x),P_1(x),P_2(x)]^\top>0$ for all $ j\in \{1,2\}$ and $0<x<\gamma _2$ ;

2. (V) $[1,w_{j}(x),w_{2}(x)^2][Q_0(x),Q_1(x),Q_2(x)]^\top>0$ for all $ j\in \{1,2\}$ and $0<x<\gamma _2$ .

Both can now be verified using Sturm’s method.

Step 8. The proof that $\widehat H(x)>0$ for $c<|x|^2 < a-2$ is more involved. Let $v(w) = -{w^2 \psi _S}+\varphi $ and $\Delta ^{-l/2}=\sum _{n\geq -l} \delta _{l,n} r^n$ . For $s=|x|^2>a-2$ , we have

$$ \begin{align*} \frac{\widehat H(x)}{\sin^2(\tfrac{\pi}{2} |x|^2)} & = \int_1^{\infty } \frac{v(\pi t)}{\Delta^{l/2}(it)}e^{- \pi t s}\mathrm{d} t + \int_1^{\infty } \frac{-\pi^2\psi(it)-\varphi_S(it)}{\Delta^{l/2}(it)}e^{- \pi |x|^2/t} t^{-d/2}\mathrm{d} t \\ & = \int_1^{\infty } v(\pi t)\bigg(\sum_{n=-l}^{N} \delta_{l,n} e^{-n\pi t}\bigg)e^{- \pi t s}\mathrm{d} t + \int_1^{\infty } v(\pi t)\bigg(\sum_{n>N} \delta_{l,n} e^{-n\pi t}\bigg)e^{- \pi t s}\mathrm{d} t \\ & \quad + \int_1^{\infty } \frac{-\pi^2\psi(it)-\varphi_S(it)}{\Delta^{l/2}(it)}e^{- \pi s/t} t^{-d/2}\mathrm{d} t. \end{align*} $$

In this way, the last two integrals above converge absolutely for $s=|x|^2>0$ , while the first integral extends to a meromorphic function of $s\in \mathbb {C}$ with possible poles $s=a-2,a-4,\ldots ,2,0$ . Since $\widehat H(x)$ is entire in the variable s, the above representation now holds in the region $\Re s \in (0,\infty )\setminus \{2,4,\ldots ,,a-2\}$ . In particular, since $\delta _{l,n}\geq 0$ , $v(\pi t)>0$ and $-\pi ^2\psi (it)-\varphi _S(it)>0$ for $t\geq 1$ , we obtain the following inequality in the range $s>0$ :

$$ \begin{align*}\frac{\widehat H(x)}{\sin^2(\tfrac{\pi}{2} |x|^2)}> \widetilde{ \int}_{1}^\infty {v(\pi t)}\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) e^{-\pi t s} \mathrm{d} t , \end{align*} $$

where by $ \widetilde { \int }$ we mean the meromorphic extension of the function defined by this integral. Now let $A(w)$ be such that $v(w)-A(w)=O(r^l)$ . Then the right-hand side above is

$$ \begin{align*} & \widetilde{ \int}_{1}^\infty A(\pi t)\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) e^{-\pi t s} \mathrm{d} t + { \int}_{1}^\infty (v(\pi t)-A(\pi t))\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) e^{-\pi t s} \mathrm{d} t \\ &> \widetilde{ \int}_{1}^\infty A(\pi t)\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) e^{-\pi t s} \mathrm{d} t \\ & \quad \quad + { \int}_{1}^\infty (-{(w^2 \psi_S)^{\mathrm{trunc}}}(it)+{(\varphi)_{\mathrm{trunc}}}(it)-A(\pi t ))\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) e^{-\pi t s} \mathrm{d} t \\ & = \widetilde{ \int}_{1}^\infty (-{(w^2 \psi_S)^{\mathrm{trunc}}}(it)+{(\varphi)_{\mathrm{trunc}}}(it))\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) e^{-\pi t s} \mathrm{d} t, \end{align*} $$

for $s>0$ . After a change of variables $w=\pi t$ , we conclude that

$$ \begin{align*} \frac{\pi \widehat H(x)}{\sin^2(\tfrac{\pi}{2} |x|^2)}> \widetilde{\int}_{\pi}^\infty \bigg( \sum_{m=-a+2}^{2N-l-2} p_m(w)e^{-mw} \bigg) e^{-w s}\mathrm{d} w \end{align*} $$

for $s>0$ , where

$$ \begin{align*}(-{(w^2 \psi_S)^{\mathrm{trunc}}}(it)+{(\varphi)_{\mathrm{trunc}}}(it))\bigg(\sum_{n= -l}^{N-l-2} \delta_{l,n} e^{-n\pi t} \bigg) =: \sum_{m=-a+2}^{2N-l-2} p_m(w)e^{-mw}. \end{align*} $$

Let now $M=\lfloor (2N-l-2)/2^{m}\rfloor $ with $m=4,3,2,1,0$ (depending on precision). Let

$$ \begin{align*} A_1(w)&=\sum_{m=-a+2}^{M-1} p_m(w)e^{-mw} + e^{-Mw}w^2[w^2](p_M(w)) \\ A_2(w)&=-A_1(w)+\sum_{m=-a+2}^{2N-l-2} p_m(w)e^{-mw}. \end{align*} $$

Let $x=e^{-w}$ . We then show that

1. (VI) $[1,w_{j}(x),w_{2}(x)^2][[w^0]A_2,[w^1]A_2,[w^2]A_2]^\top>0$ for all $ j\in \{1,2\}$ and $0<x<\gamma _2$ .

This proves that

(4) $$ \begin{align} \frac{\pi \widehat H(x)}{\sin^2(\tfrac{\pi}{2} |x|^2)}> \widetilde{\int}_{\pi}^\infty A_1(w) e^{-w s}\mathrm{d} w. \end{align} $$

Let now $\mathcal {R}$ be the following ‘rationalization’ operator

$$ \begin{align*}P(s)=\sum c_{i,j,n} \pi^i e^{-\pi j} s^n \mapsto \mathcal{R}(P)(s):= \sum \widetilde c_{i,j,n} s^n, \end{align*} $$

where $\widetilde c_{i,j,n}=\min _{\alpha ,\beta \in \{1,2\}}\{c_{i,j,n} \pi _\alpha ^i \gamma _\beta ^j\}$ and $\pi _1,\pi _2,\gamma _1,\gamma _2$ are rational approximations of $\pi $ and $\gamma =e^{-\pi }$ such that

$$ \begin{align*}\pi_1 < \pi < \pi_2 \quad \text{and} \quad \gamma_1 < \gamma<\gamma_2. \end{align*} $$

Usually these rational approximations are taken to be m-digit truncations (in base $10$ ) from below and above, with $m\in \{10,15,20,\ldots ,50\}$ depending on the required precision. Observe now that if $p(w)$ is a quadratic polynomial with coefficients in $\mathbb {Q}[\pi ]$ , then we obtain that

$$ \begin{align*} & e^{\pi s} \int_\pi^\infty p(w)e^{-mw}e^{-sw} \mathrm{d} w \\ & = e^{-\pi m}\int_0^\infty p(w+\pi)e^{-(s+m)w} \mathrm{d} w \\ & = e^{-\pi m} \sum_{j=0}^2 \frac{j)}{(s+m)^{j+1}} \\ & = e^{-\pi m} \frac{(s+m)^3[w^0](p(w+\pi))+(s+m)^2[w^1](p(w+\pi))+2(s+m)[w^2](p(w+\pi))}{(s+m)^4} \\ & \geq \frac{\mathcal{R}(e^{-\pi m}(s+m)[w^0](p(w+\pi)))}{(s+m)^2}+\frac{\mathcal{R}(e^{-\pi m}[w^1](p(w+\pi)))}{(s+m)^2}\\ &\quad +\frac{\mathcal{R}(2e^{-\pi m}(s+m)[w^2](p(w+\pi)))}{(s+m)^4}\\ & =: {B_m[p](s)}. \end{align*} $$

We deduce that (4) is bounded from below by

$$ \begin{align*} e^{-\pi s}\bigg(\sum_{m=-a+2}^{M-1} B_m(p_m)(s) + B_M(w^2[w^2](p_M(w))) \bigg) =: e^{-\pi s}Q(s), \end{align*} $$

where $Q(s)$ is a rational function with rational coefficients. Finally, we write $Q=Q_{num}/Q_{den}$ , for polynomials $Q_{num}$ and $Q_{den}$ and obtain that $\widehat H(x)>0$ for $c<|x|^2 < a-2$ holds true if the following condition is satisfied

1. (VII) $Q_{den}(s) \prod _{j=0}^{a/2-1} (s-2j)^{-2}$ has only nonnegative coefficients and $Q_{num}(s)>0$ for $c<s<a-2$ .

This can be checked by Sturm’s method again. Notice that since we have used only strict inequalities, this shows that $\{|x|^2 : \widehat H(x)=0 \text { and } |x|>c_d\}=\{a_d,a_d+2,\ldots \}$ .

We then check that conditions (I), (II), (III), (IV), (V), (VI) and (VII) are satisfied using rational arithmetic only, producing in this way a mathematical proof. The necessary algorithm to check this positivity conditions can be found in the ancillary files in the arXiv submission of this paper (a file named Postest; please also read the file Readme).

Step 9. Finally, for $d=48$ , it remains to show the claim that $\{|x|^2:H(x)< 0\}\cap (0,10) = (6,8)$ . The method is exactly the same as the one employed on Step 8, except we start from $H(x)$ and $-{w^2 \psi _S}-\varphi $ and show, after essentially the same procedure, that the resulting rational function $Q(s)$ divided by $(s-6)(s-8)$ is positive in the interval $0<s<10$ .

This finishes the proof of Theorem 4.

9 Proof of Theorems 2 and 5

Noting that $A_d = (1,\sup (K_d)]\setminus K_d$ , we conclude that a packing P is K-admissible if and only if it avoids $A_d$ . Thus, Theorems 2 and 5 are equivalent. The proof of Theorem 5 follows directly from Theorem 4. Let $P=\Lambda +Y+{\tfrac 12}B_d$ be a $K_d$ -admissible periodic sphere packing. Poisson summation over $\Lambda +Y-Y$ with the function $F(x)=H(\sqrt {a_d}x)$ and H as in Theorem 4 shows that

$$ \begin{align*}H(0)\#Y \geq \sum_{y,y'\in Y} \sum_{x\in L} F(x+y-y') = \frac{1}{\mathrm{vol}(\mathbb{R}^d/L)} \sum_{x^*\in L^*} \widehat F(x^*)\bigg|\sum_{y\in Y} e^{2\pi i y \cdot x^*}\bigg|^2 \geq \frac{\widehat H(0)a_d^{-d/2}(\#Y)^2}{\mathrm{vol}(\mathbb{R}^d/L)}; \end{align*} $$

hence, $\mathrm {dens}(P)\leq \mathrm {vol} \left ( B_d\right ) \left ( \frac {\sqrt {a_d}}{2}\right )^d$ , which is attained by any even unimodular extremal lattice. Poisson summation implies that equality is attained for a lattice packing ( $\#Y=1$ ) if and only if for every $v\in \Lambda ^*$ , we have $|v|^2\in \{a_d,a_d+2,\ldots \}$ . Then, as in the proof of Theorem 1, one shows that $\sqrt {a_d}\Lambda $ is an even unimodular extremal lattice.