Schmidt's Game and Vitali Sets

Abstract

While many types of non-measurable sets are never $(\hat{I}\pm ,{\hat{I}}^2)$-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 $\hat{I}\pm ,{\hat{I}}^2$ one can construct a Vitali set which is $(\hat{I}\pm ,{\hat{I}}^2)$-winning, while for other values of $\hat{I}\pm ,{\hat{I}}^2$ every Vitali set is $(\hat{I}\pm ,{\hat{I}}^2)$-losing. We also investigate the $(\hat{I}\pm ,{\hat{I}}^2)$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets.

9 pages, 3 figures