A Note on Lower Bounds in Szemerédi's Theorem with Random Differences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909718621978624 |
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| author | Zheng, Jason |
| author_facet | Zheng, Jason |
| contents | In this note, we consider Szemerédi's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a generalization of an argument of Altman with Moshkovitz--Zhu's bounds for the partition rank of a tensor in terms of its analytic rank, we (slightly) improve the best known lower bounds (due to Briët) on the size $|S|$ required for Szemerédi's theorem with difference in $S$ to hold asymptotically almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on Lower Bounds in Szemerédi's Theorem with Random Differences Zheng, Jason Number Theory Combinatorics 11B30 In this note, we consider Szemerédi's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a generalization of an argument of Altman with Moshkovitz--Zhu's bounds for the partition rank of a tensor in terms of its analytic rank, we (slightly) improve the best known lower bounds (due to Briët) on the size $|S|$ required for Szemerédi's theorem with difference in $S$ to hold asymptotically almost surely. |
| title | A Note on Lower Bounds in Szemerédi's Theorem with Random Differences |
| topic | Number Theory Combinatorics 11B30 |
| url | https://arxiv.org/abs/2508.01187 |