Hausdorff dimension of differences of badly approximable sets

TL;DR

This paper proves that the difference between the sets of badly approximable numbers and inhomogeneously badly approximable numbers also has full Hausdorff dimension, using a novel variant of Schmidt's game called the rapid game.

Contribution

It introduces the rapid game, a new variant of Schmidt's game, to establish the full dimension of the difference set of badly approximable numbers.

Findings

The difference set of badly approximable numbers has full Hausdorff dimension.

The rapid game effectively analyzes the dimension of complex sets in Diophantine approximation.

The method extends the understanding of the structure of badly approximable sets.

Abstract

The set of badly approximable numbers, Bad, is known to be winning for Schmidt's game and hence has full Hausdorff dimension. It is also known that the set of inhomogeneously badly approximable numbers has full dimension. We prove that the set difference also has full dimension using a variant of the Schmidt game, which we call the rapid game, played on the space of unimodular grids.