# Simon Kristensen

## 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 language English Århus Department of Mathematical Sciences, Aarhus University 1-12 12 Published - 2008

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ID: 14665164