$2$-large sets are sets of Bohr recurrence
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914218305912832 |
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| author | Alweiss, Ryan |
| author_facet | Alweiss, Ryan |
| contents | Let $α_1, \cdots, α_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||α_i s||_{\mathbb{R}/\mathbb{Z}}>δ$ for some $i$ and some fixed $δ>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is a $2$-coloring of $\mathbb{N}$ that avoids arbitrarily long arithmetic progressions with common differences in $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $2$-large sets are sets of Bohr recurrence Alweiss, Ryan Combinatorics Dynamical Systems Number Theory Let $α_1, \cdots, α_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||α_i s||_{\mathbb{R}/\mathbb{Z}}>δ$ for some $i$ and some fixed $δ>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is a $2$-coloring of $\mathbb{N}$ that avoids arbitrarily long arithmetic progressions with common differences in $S$. |
| title | $2$-large sets are sets of Bohr recurrence |
| topic | Combinatorics Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2512.01997 |