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 a₁x₁+ +aₖxₖ=0 with k≥ 4 has at most align* & exp(-c(log N)1/5+o(1))Nalign* 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.