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.

@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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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}}