Invariant Equations in Many Variables

We show that if a set does not contain any non-trivial solutions to an invariant equation of length \(k\geq 4\cdot 3^{m}+2\) for some positive integer $m$, then its size is at most \(\exp(-c\log^{1/(6+\gamma_m)} N)N\), where \(\gamma_m = 2^{-m}\). We prove a lower bound of \(\exp(-C\log^7(2/\alpha))N^{k-1}\) to the number of solutions of an invariant equation in \(k\geq 4\) variables, contained in a set of density \(\alpha\). To compliment that result in the case of convex equations, we give a Behrend-type construction for the same problem with the number of solutions of a convex equation bounded above by \(\exp(-c\log^2(2/\alpha))N^{k-1}\).