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 C, where B and C are disjoint sets satisfying \ aligned |\ (b₁, , b₂ₒ) B^{2s₈=₁ˢ bᵢ &=₈=₁ˢ bₒ+₈\}|ₒ|B|^2s - (s) ^{1/2 - o (1) }, \\ |\ (c₁, , c₂ₒ) C^{2s₈=₁ˢ cᵢ&=₈=₁ˢ cₒ+₈\}|ₒ|C|^2s - (s) ^{1/2 - o (1) }. aligned \ 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. A key ingredient in our proofs is an optimal variant of the s-fold Balog--Szemer\'edi--Gowers theorem.