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arXiv · 2608.17365

A Counting Lemma for Somewhat Restricted 3-APs

Amey Bhangale·Subhash Khot·Yang P. Liu·Dor Minzer
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Abstract

For a prime $p\geq 3$, a somewhat restricted $3$-AP in $\mathbb{F}_p^n$ is a triplet $(x,x+a,x+2a)$, where $x\in\mathbb{F}_p^n$ and $a\in \{0,1,2\}^n$. We prove a counting lemma for somewhat restricted $3$-APs in dense sets in $\mathbb{F}_p^n$. More precisely, we prove that for all $\alpha>0$, there exists $\beta>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at least $\alpha$, then it contains at least $\beta$ fraction of all somewhat restricted $3$-APs. Our proof builds on recently developed machinery from [Bhangale, Khot, Minzer, 2026]. Our main new ingredient is an arithmetic regularity lemma for patterns such as somewhat restricted 3-APs. This result is in the spirit of arithmetic regularity lemmas from the theory of Gowers uniformity norms [Green, Tao, 2010] and may be of independent interest.

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BibTeXRIS

Amey Bhangale, Subhash Khot, Yang P. Liu, Dor Minzer. 2026-08-18. A Counting Lemma for Somewhat Restricted 3-APs. https://arxiv.org/abs/2608.17365

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