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arXiv · 2508.01187

A Note on Lower Bounds in Szemer\'edi's Theorem with Random Differences

Jason Zheng
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Abstract

In this note, we consider Szemer\'{e}di'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 Bri\"et) on the size $|S|$ required for Szemer\'edi's theorem with difference in $S$ to hold asymptotically almost surely.

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Jason Zheng. 2025-08-02. A Note on Lower Bounds in Szemer\'edi's Theorem with Random Differences. https://arxiv.org/abs/2508.01187

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