  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.
