Roth-type theorems in $K_{s,t}$-free sets

TL;DR

This paper proves that large $K_{s,t}$-free subsets of integers and vector spaces necessarily contain solutions to all fixed translation-invariant linear equations with at least five variables, extending previous results.

Contribution

It extends earlier results on Sidon sets to the full family of $K_{s,t}$-free sets and provides stronger bounds in vector spaces using Fourier analysis and polynomial methods.

Findings

Large $K_{s,t}$-free sets contain solutions to all fixed translation-invariant linear equations with ≥5 variables.

In vector spaces, stronger bounds including polylogarithmic savings are achieved.

The results generalize previous work on Sidon sets to broader classes of $K_{s,t}$-free sets.

Abstract

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $\Omega(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets. We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lov\'asz-Sauermann.