Energy estimates in sum-product and convexity problems

Abstract

We prove a new class of low-energy decompositions which, amongst other consequences, imply that any finite set $A$ of integers may be written as $A = B \cup C$, where $B$ and $C$ are disjoint sets satisfying \[ |\{ (b_1, \dots, b_{2s}) \in B^{2s} \ | \ b_1 + \dots + b_{s} = b_{s+1} + \dots + b_{2s}\}| \ll_{s} |B|^{2s - (\log \log s)^{1/2 - o(1)}} \] and \[ |\{ (c_1, \dots, c_{2s}) \in C^{2s} \ | \ c_1 \dots c_{s} = c_{s+1} \dots c_{2s} \}| \ll_{s} |C|^{2s - (\log \log s)^{1/2 - o(1)}}.\] This generalises previous results of Bourgain--Chang on many-fold sumsets and product sets to the setting of many-fold energies, albeit with a weaker power saving, consequently confirming a speculation of Balog--Wooley. We further use our method to obtain new estimates for $s$-fold additive energies of $k$-convex sets, and these come arbitrarily close to the known lower bounds as $s$ becomes sufficiently large.