On shrinking targets for Z^m actions on tori

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.

References in corpus (2)

• Classical metric Diophantine approximation revisited
• Remarks on shrinking target properties

Cited by in corpus (7)

• Schmidt's game, fractals, and orbits of toral endomorphisms
• Badly approximable systems of affine forms, fractals, and Schmidt games
• A note on badly approximable linear forms on manifolds
• Badly approximable vectors on rational quadratic varieties
• Inhomogeneous Diophantine approximation in the coprime setting
• Schmidt's game, fractals, and numbers normal to no base
• Badly approximable affine forms and Schmidt games