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 $O(1/\log^{ *} n)$, which follows from the density Hales-Jewett theorem.
Categories: math.CO cs.DM