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
is a natural entropic analogue of the additive energy
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–Szemerédi–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.
