Entropic Additive Energy and Entropy Inequalities for Sums and Products

Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums, products, and sum-product combinations of continuous random variables. Partly motivated by recent work by Goh on the discrete entropic version of the notion of “additive energy”, we introduce the additive energy of pairs of continuous random variables and prove various versions of the statement that “the additive energy is large if and only if the entropy of the sum is small”, along with a version of the Balog–Szemerédi–Gowers theorem for differential entropy. Then, motivated in part by recent work by M´athé and O’Regan, we establish a series of new differential entropy inequalities for products and sum-product combinations of continuous random variables. In particular, we prove a new, general, ring Plüunnecke–Ruzsa entropy inequality. We briefly return to the case of discrete entropy and provide a characterization of discrete random variables with “large doubling”, analogous to Tao’s Freiman-type inverse sumset theory for the case of small doubling. Finally, we consider the natural entropic analog of the Erdős–Szemerédi sum-product phenomenon for integer-valued random variables. We show that, if it does hold, then the range of parameters for which it does would necessarily be significantly more restricted than its anticipated combinatorial counterpart.