Translation invariant equations and the method of Sanders

TL;DR

This paper extends recent advances in additive combinatorics to find solutions to translation invariant linear equations in various algebraic structures, improving understanding of their solution sets.

Contribution

It generalizes Sanders' methods to broader settings, providing new results for linear equations in cyclic groups and polynomial rings.

Findings

Improved bounds for solutions to translation invariant equations

Extension of Sanders' techniques to new algebraic contexts

Enhanced understanding of solution structures in finite fields and cyclic groups

Abstract

We extend the recent improvement of Roth's theorem on three term arithmetic progressions by Sanders to obtain similar results for the problem of locating non-trivial solutions to translation invariant linear equations in many variables in both $\bbz/N\bbz$ and $\bbf_q[t]$.