On the polynomial Szemerédi theorem in finite fields

Abstract

Let P 1 , . . . , P m ∈ Z[y] be any linearly independent polynomials with zero constant term. We show that there exists γ > 0 such that any subset of F q of size at least q 1−γ contains a nontrivial polynomial progression x, x + P 1 (y), . . . , x + P m (y), provided the characteristic of F q is large enough.

Bounds for sets with no polynomial progressions

“…The proof of Theorem 1.1 proceeds via a density increment argument where, as in [15], it is shown that any subset of [ ] with no nontrivial polynomial progressions has increased density on a long arithmetic progression with very small common difference. This is done by following the strategy for proving quantitative bounds in the polynomial Szemerédi theorem originating in [14], which is to first show that the count of polynomial progressions in a set is controlled by some Gowers -norm and then to show that, in certain situations, one can combine this -control with understanding of shorter progressions to deduce −1 -control. We refer to this second part of the argument as a 'degree-lowering' result, and it is here that it is crucial that 1 , .…”

“…In contrast to the finite field situation of [14], the main challenge in this artice is to first prove that the count of polynomial progressions is controlled by some -norm. By using repeated applications of the van der Corput inequality following Bergelson and Leibman's [2] PET induction scheme, we can prove control in terms of an average of a certain family of Gowers box norms.…”

“…In [15], the author and Prendiville adapted the degree-lowering method of [14] to handle the progression , + , + 2 in the integer setting. The adaptation in that paper quickly breaks down for essentially all other nonlinear progressions, however.…”

“…Theorem 1.1 thus brings our knowledge of the polynomial Szemerédi theorem in the integers more in line with what is known in finite fields. The proof of Theorem 1.1 involves adapting the central idea of [14] to the integer setting. Such an adaptation was first done by Prendiville and the author [15] for the special case of the progression , + , + 2 , showing that , 2 ( ) ≪ /(log log ) for some absolute constant > 0.…”

“…. , have distinct degrees in Theorem 1.1 is the exact condition needed to adapt the argument of [14] to the integers in full. We will say more about why this is the case in Section 3.…”