Roth's theorem in many variables

We prove that if $A\subseteq\{1,\dots,N\}$ has no nontrivial solution to the equation $x_1 + x_2 + x_3 + x_4 + x_5 = 5y$, then $|A|\ll Ne^{-c(\log N)^{1/7}}$, $c> 0$. In view of the well-known Behrend construction, this estimate is close to best possible.