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

Let k ⩾ 3 be an integer, and let G be a finite abelian group with | G | = N , where ( N , ( k − 1)!) = 1. We write r k ( 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 r k (ℤ/ N ℤ) = o k ( N ). It is known, in fact, that the estimate r k ( G ) = o k ( N ) holds for all G . There have been many papers concerning the issue of finding quantitative bounds for r k ( G ). A result of Bourgain states that r 3 ( G ) ≪ N (log log N /log N ) 1/2 for all G . In this paper we obtain a similar bound for r 4 ( G ) in the particular case G = F n , where F is a fixed finite field with char ( F ) ≠ 2, 3 (for example, F = 5 ). We prove that r 4 ( 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 .