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arXiv · 2601.00401

Schmidt's Game and Vitali Sets

James Atchley·Lior Fishman·Stephen Jackson·Daozheng Liu·Emily Yao
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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.

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James Atchley, Lior Fishman, Stephen Jackson, Daozheng Liu, Emily Yao. 2026-01-01. Schmidt's Game and Vitali Sets. https://arxiv.org/abs/2601.00401

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