Published online by Cambridge University Press: 21 July 2017
This survey looks at some recent applications of relative entropy in additive combinatorics. Specifically, we examine to what extent entropy-increment arguments can replace or even outperform more traditional energy-increment strategies or alternative approximation arguments based on the Hahn-Banach theorem.
Entropy has a long history as a tool in combinatorics. Starting with a well-known estimate for the sum of the first few binomial coefficients, some of the classical applications include Spencer's theorem that six standard deviations suffice, which states that given n finite sets, there exists a two-colouring of the elements such that all sets have discrepancy at most; a proof of the Loomis-Whitney inequality, which gives an upper bound on the volume of an n-dimensional body in Euclidean space in terms of its (n - 1)-dimensional projections; or Radhakrishnans proof  of Bregman's theorem on the maximum permanent of a 0/1 matrix with given row sums. For a beautiful introduction to these fascinating applications, as well as an extensive annotated bibliography, see .
There are other more recent results in additive combinatorics in particular where the concept of entropy has played a crucial role. Notable examples include Fox's improvement  of the bounds in the graph removal lemma (see also , which appeared in the proof-reading stages of this article); Szegedy's information-theoretic approach  to Sidorenko's conjecture (see also the blog post  by Gowers); Tao's solution  to the Erds discrepancy problem (see the discussion  on Tao's blog).
Since the above developments appear to be well captured by discussions online, we shall not cover them in any detail here. Instead we shall focus on a particular strand of recent results in additive combinatorics that could all be described as “approximation theorems” of a certain kind.
The text naturally splits into five parts. To start with we give a very brief introduction to the concept of entropy and its variants, in particular relative entropy (also known as Kullback-Leibler divergence). In Section 3, we state and prove a rather general sparse approximation theorem due to Lee .