A Note on Lower Bounds in Szemerédi's Theorem with Random Differences

2025/08/02 by Zheng, Jason
#11B30 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.01187

Abstract

In this note, we consider Szemerédi's theorem on k-term arithmetic progressions over finite fields \mathbbF_pⁿ, 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.

Citations

• Quasi-linear relation between partition and analytic rank
• Subspaces of tensors with high analytic rank
• On Szemerédi's theorem with differences from a random set
• Random differences in Szemerédi's theorem and related results
• Finite field models in additive combinatorics

Related

• On Szemerédi's theorem with differences from a random set
• Random differences in Szemerédi's theorem and related results
• Popular progression differences in vector spaces
• New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
• Improved Bounds for Szemerédi's Theorem