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