Further bounds in the polynomial Szemer\'{e}di theorem over finite fields

TL;DR

This paper establishes new upper bounds for the size of subsets in finite fields that avoid specific polynomial progressions, advancing the understanding of polynomial Szemerédi theorem cases with complex polynomial relations.

Contribution

It provides the first known upper bounds for polynomial progressions involving non-linearly independent and non-homogeneous polynomials in finite fields.

Findings

Derived upper bounds for polynomial progression-free subsets.

Improved bounds for subsets lacking k-th power difference progressions.

Estimated the number of such progressions in arbitrary finite field subsets.

Abstract

We provide upper bounds for the size of subsets of finite fields lacking the polynomial progression $$ x, x+y, ..., x+(m-1)y, x+y^m, ..., x+y^{m+k-1}.$$ These are the first known upper bounds in the polynomial Szemer\'{e}di theorem for the case when polynomials are neither linearly independent nor homogeneous of the same degree. We moreover improve known bounds for subsets of finite fields lacking arithmetic progressions with a difference coming from the set of $k$-th power residues, i.e. configurations of the form $$ x, x+y^k, ..., x+(m-1)y^k.$$ Both results follow from an estimate of the number of such progressions in an arbitrary subset of a finite field.