A Counting Lemma for Somewhat Restricted 3-APs

Abstract

For a prime $p\ge 3$, 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.

58 pages