Decaying and non-decaying badly approximable numbers

TL;DR

This paper investigates the Hausdorff dimension of decaying and non-decaying badly approximable numbers, showing both sets have full dimension one, using a novel game combining Banach--Mazur and Schmidt's game.

Contribution

It answers a question by Y. Bugeaud by determining the Hausdorff dimensions of both sets as equal to one, employing a new game-theoretic approach.

Findings

Both sets of numbers have Hausdorff dimension one.

The proof introduces a combined Banach--Mazur and Schmidt's game.

The results resolve a question posed by Bugeaud in 2015.

Abstract

We call a badly approximable number $decaying$ if, roughly, the Lagrange constants of integer multiples of that number decay as fast as possible. In this terminology, a question of Y. Bugeaud ('15) asks to find the Hausdorff dimension of the set of decaying badly approximable numbers, and also of the set of badly approximable numbers which are not decaying. We answer both questions, showing that the Hausdorff dimensions of both sets are equal to one. Part of our proof utilizes a game which combines the Banach--Mazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).