On shrinking targets for Zm actions on tori

Yann Bugeaud; Stephen Harrap; S. Kristensen; Sanju Velani

On shrinking targets for Z^_m actions on tori

Yann Bugeaud, Stephen Harrap, S. Kristensen, Sanju Velani

Publikation: Working paper/Preprint › Working paper › Forskning

Abstract

Let $A$ be an $n \times m$ matrix with real entries. Consider the set $\mathbf{Bad}_A$ of $\mathbf{x} \in [0,1)^n$ for which there exists a constant $c(\mathbf{x})>0$ such that for any $\mathbf{q} \in \mathbb{Z}^m$ the distance between $\mathbf{x}$ and the point $\{A \mathbf{q}\}$ is at least $c(\mathbf{x}) |\mathbf{q}|^{-m/n}$ . It is shown that the intersection of $\mathbf{Bad}_A$ with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

Originalsprog	Engelsk
Udgivelsessted	Århus
Udgiver	Department of Mathematical Sciences, Aarhus University
Sider	1-12
Antal sider	12
Status	Udgivet - 2008

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Citationsformater

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AU - Velani, Sanju

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BT - On shrinking targets for Zm actions on tori

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On shrinking targets for Zm actions on tori

Abstract

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Citationsformater

On shrinking targets for Z^_m actions on tori