On an entropic analogue of additive energy

Abstract: Recent advances have linked various statements involving sumsets and cardinalities with corresponding statements involving sums of random variables and entropies. In this vein, this paper shows that the quantity $2{\bf H}{X, Y} - {\bf H}{X+Y}$ is a natural entropic analogue of the additive energy $E(A,B)$ between two sets. We develop some basic theory surrounding this quantity, and demonstrate its role in the proof of Tao's entropy variant of the Balog--Szemer\'edi--Gowers theorem. We examine the regime where entropic additive energy is small, and discuss a family of random variables related to Sidon sets. In finite fields, one can define an entropic multiplicative energy as well, and we formulate sum-product-type conjectures relating these two entropic energies.