Schmidt's Game and Vitali Sets
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917180026650624 |
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| author | Atchley, James Fishman, Lior Jackson, Stephen Liu, Daozheng Yao, Emily |
| author_facet | Atchley, James Fishman, Lior Jackson, Stephen Liu, Daozheng Yao, Emily |
| contents | While many types of non-measurable sets are never $(α, β)$-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 $α, β$ one can construct a Vitali set which is $(α, β)$-winning, while for other values of $α,β$ every Vitali set is $(α,β)$-losing. We also investigate the $(α,β)$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00401 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Schmidt's Game and Vitali Sets Atchley, James Fishman, Lior Jackson, Stephen Liu, Daozheng Yao, Emily Logic 03E05, 03E15 While many types of non-measurable sets are never $(α, β)$-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 $α, β$ one can construct a Vitali set which is $(α, β)$-winning, while for other values of $α,β$ every Vitali set is $(α,β)$-losing. We also investigate the $(α,β)$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets. |
| title | Schmidt's Game and Vitali Sets |
| topic | Logic 03E05, 03E15 |
| url | https://arxiv.org/abs/2601.00401 |