Abstract
A theorem of Glasner from 1979 shows that if \(A \subset \mathbb{T}= \mathbb{R}/\mathbb{Z}\) is infinite, then for each ϵ > 0 there exists an integer n such that nA is ϵ-dense and Berend—Peres later showed that in fact one can take n to be of the form f(m) for any non-constant f(x) ∈ ℤ[x]. Alon and Peres provided a general framework for this problem that has been used by Kelly—Lê and Dong to show that the same property holds for various linear actions on \({\mathbb{T}^d}\). We complement the result of Kelly—Lê on the ϵ-dense images of integer polynomial matrices in some subtorus of \({\mathbb{T}^d}\) by classifying those integer polynomial matrices that have the Glasner property in the full torus \({\mathbb{T}^d}\). We also extend a recent result of Dong by showing that if Γ ≤ SLd(ℤ) is generated by finitely many unipotents and acts irreducibly on ℝd, then the action \(\Gamma \curvearrowright {\mathbb{T}^d}\) has a uniform Glasner property.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Alon and Y. Peres, Uniform dilations, Geometric and Functional Analysis 2 (1992), 1–28.
J. T. Barton, H. L. Montgomery and J. D. Vaaler, Note on a Diophantine inequality in several variables, Proceedings of the American Mathematical Society 129 (2001), 337–345.
Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III), Annals of Mathematics 178 (2013), 1017–1059.
D. Berend and Y. Peres, Asymptotically dense dilations of sets on the circle, Journal of the London Mathematical Society 47 (1993), 1–17.
C. Dong, On density of infinite subsets I, Discrete and Continuous Dynamical Systems 39 (2019), 2343–2359.
C. Dong, On density of infinite subsets II: Dynamics on homogeneous spaces, Proceedings of the American Mathematical Society 147 (2019), 751–761.
S. Glasner, Almost periodic sets and measures on the torus, Israel Journal of Mathematics 32 (1979), 161–172.
L.-K. Hua, On an exponential sum, Journal of the Chinese Mathematical Society 2 (1940), 301–312.
L.-K. Hua, Additive Theory of Prime Numbers, Translations of Mathematical Monographs, Vol. 13, American Mathematical Society, Providence, RI, 1965.
M. Kelly and T. H. Lê, Uniform dilations in higher dimensions, Journal of the London Mathematical Society 88 (2013), 925–940.
Acknowledgement
The authors were partially supported by the Australian Research Council grant DP210100162.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bulinski, K., Fish, A. Glasner property for unipotently generated group actions on tori. Isr. J. Math. 255, 109–122 (2023). https://doi.org/10.1007/s11856-022-2412-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-022-2412-5