The development of the field of additive combinatorics in recent years has provided, among other things, a collection of fascinating and deep, elementary tools for estimating the sizes of discrete subsets of abelian groups. Tao in 2010 connected these results with the entropy of discrete probability measures: Interpreting the entropy of a discrete random variable as the logarithm of its “effective support size,” he provided a series of new inequalities for the discrete entropy. We will review this background and describe how Tao’s results extend in a nontrivial way to the entropy of random elements in general abelian groups. The somewhat surprising key difference between the discrete and the general case is that the “functional submodularity” property of the discrete entropy needs to be replaced by the general “data processing property” of the entropy.
No background in information theory, entropy or additive combinatorics will be assumed.