$\delta$-Badly approximable numbers and ubiquitously losing sets

TL;DR

This paper studies the structure and size of certain sets of badly approximable numbers, introducing new concepts like ubiquitously losing sets and establishing their winning properties and Hausdorff dimension bounds.

Contribution

It introduces the notion of ubiquitously losing sets within Schmidt game theory and provides bounds on their Hausdorff dimension, advancing understanding of badly approximable sets.

Findings

$oldsymbol{ ext{Bad}}(oldsymbol{ ext{delta}})$ is a $(1/3, 18 oldsymbol{ ext{delta}})$-winning set.

Provides a lower bound on the Hausdorff dimension of $oldsymbol{ ext{Bad}}(oldsymbol{ ext{delta}})$.

Shows $oldsymbol{ ext{Bad}}(oldsymbol{ ext{delta}})$ is a $(1/2, 18/oldsymbol{ ext{delta}})$-ubiquitously losing set.

Abstract

We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(\delta) \subset \boldsymbol{\operatorname{Bad}}(\delta')$ for $\delta \geq \delta'>0$ on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the $\tau$-well approximable numbers. We show that the set $\boldsymbol{\operatorname{Bad}}(\delta)$ is a $(1/3, 18 \delta)$-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of $(\alpha, \beta)$-$\textit{ubiquitously losing sets}$ to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an $(\alpha, \beta)$-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that $\boldsymbol{\operatorname{Bad}}(\delta)$ is a $(1/2, 18/\delta)$-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of $\boldsymbol{\operatorname{Bad}}(\delta)$ that is strictly less than one. Combined with a finite intersection property and a bilipschitz transfer property, we obtain results for finite intersections of translates of $\boldsymbol{\operatorname{Bad}}(\delta)$.