Badly approximable affine forms and Schmidt games

TL;DR

This paper demonstrates that the set of real numbers with a specific badly approximable property related to affine forms is winning in the sense of Schmidt games, indicating its large size and robustness.

Contribution

It establishes that the set of numbers satisfying a certain affine approximation condition is 1/8-winning, extending the understanding of badly approximable sets in Diophantine approximation.

Findings

The set is 1/8-winning in Schmidt games.

The set has full Hausdorff dimension.

The result applies to affine forms with a specific approximation property.

Abstract

For any real number \t, the set of all real numbers x for which there exists a constant c(x) > 0 such that \inf_{p \in \ZZ} |\t q - x - p| \geq c(x)/|q| for all q in \ZZ {0} is an 1/8-winning set.