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On shrinking targets for Zm actions on tori

Research output: Working paper/Preprint Working paperResearch

  • Yann Bugeaud, Université Louis Pasteur, France
  • Stephen Harrap, University of York, United Kingdom
  • S. Kristensen
  • Sanju Velani, University of York, United Kingdom
  • Department of Mathematical Sciences
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.
Original languageEnglish
Place of publicationÅrhus
PublisherDepartment of Mathematical Sciences, Aarhus University
Number of pages12
Publication statusPublished - 2008

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