Energy estimates in sum-product and convexity problems

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 \[ {aligned} |\{(b_1,,b₂ₛ)∈ B²ˢ∑ᵢ₌₁^s b_i &=∑ᵢ₌₁^s bₛ₊ᵢ\}|ₛ|B|^{2s - (loglog s)1/2 - o(1)},\\ |\{(c_1,,c₂ₛ)∈ C²ˢ∏ᵢ₌₁^s c_i&=∏ᵢ₌₁^s cₛ₊ᵢ\}|ₛ|C|^{2s - (loglog s)1/2 - o(1)}. {aligned} \] 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.