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Equidistribution in the complex plane and self-similar measures
Part of:
Probabilistic theory: distribution modulo $1$; metric theory of algorithms
Classical measure theory
Published online by Cambridge University Press: 09 September 2025
Abstract
We establish the pointwise equidistribution of self-similar measures in the complex plane. Let 
$\beta \in \mathbb Z[\mathrm{i}]$, whose complex conjugate 
$\overline{\beta}$ is not a divisor of β, and 
$T \subset \mathbb Z[\mathrm{i}]$ a finite subset. Let µ be a non-atomic self-similar measure with respect to the IFS 
$\big\{f_{t}(z)=\frac{z+t}{\beta}\colon t\in T\big\}$. For 
$\alpha \in \mathbb Z[\mathrm{i}]$, if α and β are relatively prime, then we show that the sequence 
$(\alpha^n z)_{n\ge 1}$ is equidistributed modulo one for µ-almost everywhere 
$z \in \mathbb{C}$. We also discuss normality of radix expansions in Gaussian integer base, and obtain pointwise normality. Our results generalize partially the classical results in the real line to the complex plane.
Keywords
MSC classification
Primary:
11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary:
28A80: Fractals
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.
References
Algom, A.. A simultaneous version of Host’s equidistribution theorem. Trans. Amer. Math. Soc. 373 (2020), 8439–8462.CrossRefGoogle Scholar
Algom, A.. Actions of diagonal endomorphisms on conformally invariant measures on the 2-torus. Monatsh. Math. 195 (2021), 545–564.CrossRefGoogle Scholar
Algom, A.. Recent progress on pointwise normality of self-similar measures. (arXiv:2504.18192), 2025.Google Scholar
Algom, A., Baker, S. and Shmerkin, P.. On normal numbers and self-similar measures. Adv. Math. 399 (2022), .CrossRefGoogle Scholar
Algom, A., Rodriguez Hertz, F. and Wang, Z.. Pointwise normality and Fourier decay for self-conformal measures. Adv. Math. 393 (2021), .CrossRefGoogle Scholar
Algom, A., Rodriguez Hertz, F. and Wang, Z.. Spectral gaps and Fourier decay for self-conformal measures in the plane. (arXiv:2407.11688), 2024.Google Scholar
Bárány, B., Käenmäki, A., Pyörälä, A. and Wu, M.. Scaling limits of self-conformal measures. (arXiv:2308.11399), 2023.Google Scholar
Baker, S., Khalil, O. and Sahlsten, T.. Fourier decay from l 2-flattening. (arXiv:2407.16699), 2024.Google Scholar
Bisbas, A.. Normal numbers from infinite convolution measures. Ergodic Theory Dynam. Systems. 23 (2003), 389–393.CrossRefGoogle Scholar
Bisbas, A.. A Cassels-Schmidt theorem for non-homogeneous Markov chains. Bull. Sci. Math. 129 (2005), 25–37.CrossRefGoogle Scholar
Borel, E.. Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo (1884-1940). 27 (1909), 247–271.CrossRefGoogle Scholar
Brown, G., Moran, W. and Pearce, C. E. M.. Riesz products and normal numbers. J. London Math. Soc. 32 (1985), 12–18.CrossRefGoogle Scholar
Brown, G., Moran, W. and Pearce, C. E. M.. Riesz products, Hausdorff dimension and normal numbers. Math. Proc. Cambridge Philos. Soc. 101 (1987), 529–540.CrossRefGoogle Scholar
Brown, G., Moran, W. and Pollington, A. D.. Normality to noninteger bases. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 1241–1244.Google Scholar
Cassels, J. W. S.. On a problem of Steinhaus about normal numbers. Colloq. Math. 7 (1959), 95–101.CrossRefGoogle Scholar
Davenport, H., Erdos, P. and LeVeque, W. J.. On Weyl’s criterion for uniform distribution. Michigan Math. J. 10 (1963), 311–314.CrossRefGoogle Scholar
Dayan, Y., Ganguly, A. and Weiss, B.. Random walks on tori and normal numbers in self-similar sets. Amer. J. Math. 146 (2024), 467–493.CrossRefGoogle Scholar
Drmota, M. and Tichy, R. F.. Sequences, Discrepancies and applications (Lecture Notes in Mathematics). Vol.1651, (Springer-Verlag, 1997).CrossRefGoogle Scholar
Feldman, J. and Smorodinsky, M.. Normal numbers from independent processes. Ergodic Theory Dynam. Systems. 12 (1992), 707–712.CrossRefGoogle Scholar
Gao, X., Ma, J., Song, K. and Zhang, Y.. On the Fourier transform of coin-tossing type measures. J. Math. Anal. Appl. 484 (2020), :123706.CrossRefGoogle Scholar
Hochman, M.. A short proof of Host’s equidistribution theorem. Israel J. Math. 251 (2022), 527–539.CrossRefGoogle Scholar
Hochman, M. and Shmerkin, P.. Equidistribution from fractal measures. Invent. Math. 202 (2015), 427–479.CrossRefGoogle Scholar
Host, B.. Nombres normaux, entropie, translations. Israel J. Math. 91 (1995), 419–428.CrossRefGoogle Scholar
Host, B.. Some results of uniform distribution in the multidimensional torus. Ergodic Theory Dynam. Systems. 20 (2000), 439–452.CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
Kátai, I and Szabó, J.. Canonical number systems for complex integers. Acta Sci. Math. 37 (1975), 255–260.Google Scholar
Lagarias, J. C. and Wang, Y.. Self-affine tiles in 
$ \lt b \gt R \lt /b \gt ^n$. Adv. Math. 121 (1996), 21–49.CrossRefGoogle Scholar
$ \lt b \gt R \lt /b \gt ^n$. Adv. Math. 121 (1996), 21–49.CrossRefGoogle ScholarLi, J. and Sahlsten, T.. Trigonometric series and self-similar sets. J. Eur. Math. Soc. 24 (2022), 341–368.CrossRefGoogle Scholar
Lindenstrauss, E.. p-adic foliation and equidistribution. Israel J. Math. 122 (2001), 29–42.CrossRefGoogle Scholar
Meiri, D.. Entropy and uniform distribution of orbits in
$ \lt b \gt T \lt /b \gt ^d$. Israel J. Math. 105 (1998), 155–183.CrossRefGoogle Scholar
$ \lt b \gt T \lt /b \gt ^d$. Israel J. Math. 105 (1998), 155–183.CrossRefGoogle ScholarMeiri, D. and Peres, Y.. Bi-invariant sets and measures have integer Hausdorff dimension. Ergodic Theory Dynam. Systems. 19 (1999), 523–534.CrossRefGoogle Scholar
Moran, W. and Pollington, A. D.. Metrical results on normality to distinct bases. J. Number Theory. 54 (1995), 180–189.CrossRefGoogle Scholar
Sahlsten, T.. Fourier transforms and iterated function systems, Recent Developments in Fractals and Related Fields: Conference on Fractals and Related Fields IV (ile de Porquerolles, France, 2022), (Springer Nature, 2025), .CrossRefGoogle Scholar
Varjú, P. and Yu, H.. Fourier decay of self-similar measures and self-similar sets of uniqueness. Anal. PDE. 15 (2022), 843–858.CrossRefGoogle Scholar
Vince, A.. Replicating tessellations. SIAM J. Discrete Math. 6 (1993), 501–521.CrossRefGoogle Scholar
Weyl, H.. Über ein problem aus dem gebiet der diophantischen approximationen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. (1914), 234–244.Google Scholar
Weyl, H.. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar
Xiong, Y. and Zhao, J.. Equidistribution for measures defined by digit restrictions. Nonlinearity. 35 (2022), 4064–4084.CrossRefGoogle Scholar