Effective Bounds for Restricted $3$-Arithmetic Progressions in ${\mathbb{F}}_p^n$

Abstract

For a prime $p$, a restricted arithmetic progression in ${\mathbb{F}}_p^n$ is a triplet of vectors $x,x+a,x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2{\}}^n$. What is the size of the largest $A\subseteq {\mathbb{F}}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log \log \log n{)}^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/{\log }^{\ast }n)$, which follows from the density Hales-Jewett theorem.