Around 15 years ago, a very nice connection between additive combinatorics and entropy was identified and explored primarily by Imre Ruzsa, Terence Tao and Van H. Vu. First, we are going to describe this connection and mention the underlyig information-theoretic ideas. Next, we are going to discuss a related conjecture due to Tao (2010) and draw a connection between this conjecture and the question of monotonicity in a recently established, discrete version of the entropic central limit theorem. Finally, we are going to present a recent proof of the above conjecture for the special case of log-concave random variables on the integers and discuss possible future directions.
