A Counting Lemma for Somewhat Restricted 3-APs

2026-08-18

AI Summary

本文解决了在高密度集合中计数某种程度受限的3-APs问题，通过引入一种新的算术正则性引理方法来证明存在一个下界。

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.

Problem

• somewhat restricted 3-APs
• counting lemma
• dense sets
• mathbb{F}_p^n

Innovation

• somewhat restricted 3-APs
• counting lemma
• arithmetic regularity lemma
• Gowers uniformity norms