Badly approximable systems of affine forms, fractals, and Schmidt games

Abstract

A badly approximable system of affine forms is determined by a matrix and a vector. We show Kleinbock's conjecture for badly approximable systems of affine forms: for any fixed vector, the set of badly approximable systems of affine forms is winning (in the sense of Schmidt games) even when restricted to a fractal (from a certain large class of fractals). In addition, we consider fixing the matrix instead of the vector where an analog statement holds.

14 pages, typos corrected in the current version

References in corpus (4)

• On shrinking targets for Z^m actions on tori
• A solution to a problem of Cassels and Diophantine properties of cubic numbers
• Modified Schmidt games and Diophantine approximation with weights
• Schmidt's game, Badly Approximable Linear Forms and Fractals

Cited by in corpus (2)

• Schmidt's game, fractals, and orbits of toral endomorphisms
• A note on badly approximable linear forms on manifolds