# Quantitative results using variants of Schmidt's game: Dimension bounds, arithmetic progressions, and more

## TL;DR

This paper explores quantitative aspects of Schmidt's game, particularly the potential game variant, to derive bounds on arithmetic progressions, continued fractions, and Hausdorff dimensions within fractal sets like the Cantor set.

## Contribution

It introduces the potential game as a tool for obtaining sharper quantitative bounds on properties of fractal sets, improving upon classical Schmidt's game results.

## Key findings

- Bound on the maximum length of arithmetic progressions in the middle ε-Cantor set
- Determination of the minimal n for elements with bounded continued fraction partial quotients
- Hausdorff dimension of ε-badly approximable numbers in the Cantor set

## Abstract

Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $\epsilon$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the $potential$ $game$ is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. We also use the potential game to provide a new proof of an important lemma in the classical proof of the existence of Hall's Ray.