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On shrinking targets for Z^m actions on tori

Authors
  • Bugeaud, Yann
  • Harrap, Stephen
  • Kristensen, Simon
  • Velani, Sanju
Type
Preprint
Publication Date
Dec 08, 2008
Submission Date
Jul 24, 2008
Source
arXiv
License
Yellow
External links

Abstract

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of 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.

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