A Counting Lemma for Somewhat Restricted 3-APs

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 $α>0$, there exists $β>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at least $α$, then it contains at least $β$ 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.

Disclosure

“n the spirit of rank discussed above (see Definitions 2.13 and 2.14). Finally, in Section 5 we show how to use our arithmetic regularity lemma in conjunction with the invariance principle to prove Theorem 1.4. Statement of AI use: we used ChatGPT 5.5 during the polishing of this write-up, and to write the proofs from Section A. All other mathematical content in this paper is due to the authors. 2 Preliminaries Notations. For a vector x ∈ Σn and a subset I ⊆ [n] of coordinates,”

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Drafting a complete proof for author revision

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9

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2026-08-23T05:22:01.409191Z

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