Schmidt's Game and Vitali Sets

TL;DR

This paper explores the winning properties of Vitali sets in Schmidt's game, demonstrating that some Vitali sets can be winning for certain parameters while others cannot, and compares these properties with other pathological sets.

Contribution

It establishes conditions under which Vitali sets are winning or losing in Schmidt's game, revealing nuanced behaviors not previously documented.

Findings

Certain Vitali sets are $(rac{1}{2}, rac{1}{2})$-winning.

All Vitali sets are $(rac{1}{3}, rac{1}{3})$-losing.

Different pathological sets exhibit distinct winning properties.

Abstract

While many types of non-measurable sets are never $(\alpha, \beta)$-winning in the sense of Schmidt's game, we show that this is not the case for certain Vitali sets. Our main theorems show that for certain values of $\alpha, \beta$ one can construct a Vitali set which is $(\alpha, \beta)$-winning, while for other values of $\alpha,\beta$ every Vitali set is $(\alpha,\beta)$-losing. We also investigate the $(\alpha,\beta)$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets.