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A generalization of the Kelley–Meka theorem to binary systems of linear forms

Yuval Filmus, Hamed Hatami, Kaave Hosseini, Esty Kelman

Kelley and Meka recently proved strong bounds on the size of subsets of $\mathbb{Z}_N$ or $\mathbb{F}_q^n$ that do not contain 3-term arithmetic progression. We use their techniques to prove similar bounds for subsets of $\mathbb{F}_q^n$ that do not contain non-degenerate instances of affine binary linear systems whose underlying graph is 2-degenerate.

We show that if a subset of $\mathbb{F}_q^n$ contains an atypical number of instances of an affine binary linear 2-degenerate system, then it has a constant density increment inside an affine subspace of polylogarithmic codimension. We give a counterexample showing that this kind of result does not hold for linear systems whose true complexity exceeds 1.

Using the same techniques, we obtain a counting lemma for sparse quasirandom graphs, improving on the classical result of Chung, Graham, and Wilson (Combinatorica 1989), which is only nontrivial for dense graphs.

BibTeX

@misc{FHHK23,
title = {A generalization of the {K}elley--{M}eka theorem to binary systems of linear forms},
author = {Yuval Filmus and Hamed Hatami and Kaave Hosseini and Esty Kelman},
howpublished = {arXiv:2311.12248},
year = {2023}}
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