On shrinking targets for Zm actions on tori

Research output: ResearchWorking paper

  • Yann Bugeaud
    Yann BugeaudUniversité Louis PasteurFrance
  • Stephen Harrap
    Stephen HarrapUniversity of YorkUnited Kingdom
  • S. Kristensen
  • Sanju Velani
    Sanju VelaniUniversity of YorkUnited 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
StatePublished - 2008

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