Entropy methods for sumset inequalities

Abstract

In this thesis we present several analogies betweeen sumset inequalities and entropy inequalities. We offer an overview of the different results and techniques that have been developed during the last ten years, starting with a seminal paper by Ruzsa, and also studied by authors such as Bollobás, Madiman, or Tao. After an introduction to the tools from sumset theory and entropy theory, we present and prove many sumset inequalities and their entropy analogues, with a particular emphasis on Plünnecke-type results. Functional submodularity is used to prove many of these, as well as an analogue of the Balog-Szemerédi-Gowers theorem. Partition-determined functions are used to obtain many sumset inequalities analogous to some new entropic results. Their use is generalized to other contexts, such as that of projections or polynomial compound sets. Furthermore, we present a generalization of a tool introduced by Ruzsa by extending it to a much more general setting than that of sumsets. We show how it can be used to obtain many entropy inequalities in a direct and unified way, and we extend its use to more general compound sets. Finally, we show how this device may help in finding new expanders.