On winning sets and non-normal numbers

TL;DR

This paper extends the concept of winning sets to various non-normal number sets in Cantor series expansions, demonstrating their full Hausdorff dimension and broadening understanding of their mathematical properties.

Contribution

It generalizes Schmidt's result by proving that many non-normal number sets in Cantor series expansions are winning sets, thus having full Hausdorff dimension.

Findings

Many non-normal number sets in Cantor series are winning sets.

Winning sets have full Hausdorff dimension.

Generalization of Schmidt's theorem to broader classes of expansions.

Abstract

In \cite{SchmidtGames}, W. Schmidt proved that the set of non-normal numbers in base $b$ is a {\it winning set}. We generalize this result by proving that many sets of non-normal numbers with respect to the Cantor series expansion are winning sets. As an immediate consequence, these sets will be shown to have full Hausdorff dimension.