Hausdorff dimension of differences of badly approximable sets

Abstract: The set of badly approximable numbers (real numbers with bounded continued fraction partial quotients), denoted by $\mathbf{Bad}$, has zero Lebesgue measure yet full Hausdorff dimension. In fact, it satisfies the stronger property of being a winning set in the sense of Schmidt’s game. The inhomogeneously badly approximable set, $\mathbf{Bad}^\gamma$, is also known to have full Hausdorff dimension. In this talk, we revisit these classical notions and proceed to prove that the set difference $\mathbf{Bad}^\gamma \setminus \mathbf{Bad}$ likewise has full Hausdorff dimension. Our proof relies on the dynamical interpretation of the problem on the space of unimodular grids and introduces a new variant of Schmidt’s game, the rapid game.

dynamical systemsnumber theory

Audience: researchers in the topic

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