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 ... more >>>

We give a randomized algorithmic version of the Balog--Szemer\'edi--Gowers theorem for sets of integers. Let $A\subseteq[N]$ have size $n:=|A|\geq2$, let $1\leq K\leq n$, and suppose that its additive energy satisfies $E(A)\geq n^3/K$. Reiher and Schoen proved existentially that, for every fixed $\epsilon \in (0,1/2)$, there is a subset $A'\subseteq A$ ... more >>>

The line--versus--point test asks the following local-to-global question. Suppose a function $f\colon\mathbb{F}_q^m\to\mathbb{F}_q$ is, on average over a random affine line $L$, correlated with some degree-$d$ polynomial on $L$. Must $f$ then be globally correlated with a single multivariate polynomial of degree at most $d$? Beyond being a natural combinatorial question, ... more >>>