Higher Convexity and Iterated Second Moment Estimates

Abstract

We prove bounds for the number of solutions to$$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive doubling under a convex function with sufficiently many strictly monotone derivatives. We show, roughly, that every time the number of terms in the equation is doubled, an additional saving of $1$ in the exponent of the trivial bound $N^{2k-1}$ is made, starting from the trivial case $k=1$. In the context of near-convex sets we also provide explicit dependencies on the additive doubling parameters. Higher convexity is necessary for such bounds to hold, as evinced by sets of perfect powers of consecutive integers. We exploit these stronger assumptions using a different methodology and avoiding the use of the Szemerédi-Trotter theorem, which has not been adapted to embrace higher convexity.As an application of our new estimates for $k>2$ we improve the best known bounds for sumsets of convex sets under additional convexity assumptions.