  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 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.
