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

Yifan Jing, Cosmin Pohoata, Max Wenqiang Xu

TL;DR

The paper develops a Fourier-analytic transference framework for $K_{s,t}$-free sets, establishing that dense $K_{s,t}$-free subsets of $[N]$ must contain nontrivial solutions to every fixed translation-invariant linear equation in at least five variables. It constructs dense models with controlled Fourier spectra and leverages counting lemmas to transfer dense-subset results back to sparse sets, yielding quantitative bounds. In the integers, it achieves a bound $|A|\ll N^{1-1/s}\exp(-c(\log\log N)^{1/7})$, while in finite fields $\mathbb{F}_q^n$ it obtains $|A|\ll N^{1-1/s}(\log N)^{-c}$ and polylog improvements via the arithmetic $k$-cycle removal lemma. Overall, the work extends Sidon-set type results to the full family of $K_{s,t}$-free sets and tightens the connection between extremal combinatorics and additive structure across settings.

Abstract

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $Ω(n^{1-1/s})$ 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-Lovász-Sauermann.

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

TL;DR

The paper develops a Fourier-analytic transference framework for -free sets, establishing that dense -free subsets of must contain nontrivial solutions to every fixed translation-invariant linear equation in at least five variables. It constructs dense models with controlled Fourier spectra and leverages counting lemmas to transfer dense-subset results back to sparse sets, yielding quantitative bounds. In the integers, it achieves a bound , while in finite fields it obtains and polylog improvements via the arithmetic -cycle removal lemma. Overall, the work extends Sidon-set type results to the full family of -free sets and tightens the connection between extremal combinatorics and additive structure across settings.

Abstract

We show that for all integers , any -free subset of with size 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 -free sets. We also study the corresponding problem in vector spaces over finite fields. In 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-Lovász-Sauermann.
Paper Structure (6 sections, 16 theorems, 131 equations)

This paper contains 6 sections, 16 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. $K_{s,t}$-free sets and the $s$-th moment energy
  3. A dense model theorem
  4. Counting Lemmas
  5. Proof of Theorem \ref{['JPX']}
  6. Finite field analogues for $K_{s,t}$-free sets

Key Result

Theorem 1.1

Let $k\geq 5$, and $a_1,\dots,a_k\in\mathbb{Z}\setminus\{0\}$ with $\sum_{i=1}^k a_i=0$. Let $A\subseteq [N]$ be $K_{s,t}$-free and assume that $A$ lacks nontrivial solutions of $a_1x_1+\cdots +a_kx_k=0$. Then

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 19 more