Hausdorff Dimension of Differences of Badly Approximable Sets

The set of badly approximable numbers, \(\mathbf{Bad}\), is one of the central objects in Diophantine approximation. It has Lebesgue measure zero, but full Hausdorff dimension; in fact, it is winning for Schmidt's game. One may describe \(\mathbf{Bad}\) in terms of circle rotations whose orbits avoid a shrinking neighbourhood of the origin. Replacing the origin by an arbitrary point \(\gamma\) leads to the inhomogeneous analogue \(\mathbf{Bad}^\gamma\), for which the same measure, dimension and winning results are known.

The main result of this thesis is that, for every \(\gamma \in \mathbb{R} \setminus \mathbb{Z}\), the set difference \[\mathbf{Bad}^\gamma \setminus \mathbf{Bad},\] which is not winning in the classical sense, has full Hausdorff dimension. We prove this by introducing a new variant of Schmidt's game, called the \textit{rapid game}, played on the space of unimodular grids.

Chapter 1 introduces the classical Diophantine background, including the sets \(\mathbf{Bad}\) and \(\mathbf{Bad}^\gamma\). Chapter 2 develops the required tools from the geometry of numbers. Chapter 3 turns to the dynamical background and reformulates the main problem on the space of unimodular grids. Chapter 4 introduces Schmidt-type games, in particular the rapid game, and gives a corresponding dynamical formulation of the games. In Chapter 5, the preceding material is combined to establish the main theorem. Chapter 6 sketches the higher-dimensional extension and discusses potential future directions.