Translation-Invariant Equations With at Least Four Variables

We prove that every subset of {1, ... , N } that does not contain any solutions to a translation invariant equation a( 1 )x( 1) + <middle dot> <middle dot> <middle dot> + a (k) x (k) = 0 with k >= 4 has at most exp ( - c (log N )( 1 / 5 + o ( 1 )) ) N elements, for some c > 0. This theorem improves upon previous estimates. Additionally, our method has the potential to yield an optimal estimate for this problem that matches Behrend's classical lower bound. Our approach relies on a new result on almost-periodicity of convolutions.