New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries

Abstract

Let k ⩾ 3 be an integer, and let G be a finite abelian group with |G| = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality |A| of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(ℤ/Nℤ) = ok(N). It is known, in fact, that the estimate rk(G) = ok(N) holds for all G. There have been many papers concerning the issue of finding quantitative bounds for rk(G). A result of Bourgain states thatr3(G) ≪ N(log log N/log N)1/2 for all G. In this paper we obtain a similar bound for r4(G) in the particular case G = Fn, where F is a fixed finite field with char (F) ≠ 2, 3 (for example, F = 𝔽5). We prove that r4(G) ≪ F N(log N)−c for some absolute constant c > 0. In future papers we will treat the general abelian groups G, eventually obtaining a comparable result for an arbitrary G.

Locations

• Proceedings of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• DataCite API - View
• Proceedings of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• DataCite API - View