Asymptotic upper bounds on progression-free sets in ${\mathbb{Z}}_p^n$

We show that any subset of ${\mathbb{Z}}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}<1$. In particular, we find an upper bound of $O({2.84}^n)$ on the maximum size of an affine cap in $GF(3{)}^n$.