Giorgis Petridis

New proofs of Plünnecke-type estimates for product sets in groups

We present a new method to bound the cardinality of product sets in groups and give three applications. A new and unexpectedly short proof of the Plünnecke-Ruzsa sumset inequalities for commutative groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plünnecke-Ruzsa inequalities in general groups.

The Plünnecke–Ruzsa Inequality: An Overview

In this expository article we present an overview of the Plunnecke–Ruzsa inequality: the known proofs, some of its well-known applications and possible extensions. We begin with the graph-theoretic setting in which Plunnecke and later Ruzsa worked in. The more purely combinatorial proofs of the inequality are subsequently presented. In the concluding sections we discuss the sharpness of the various results presented thus far and possible extensions of the inequality to the non-commutative setting.

The L 1-norm of exponential sums in d

Let A be a finite set of integers and F_A its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L^1-norm of F_A is at least c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L^1-norm of exponential sums of sets in the d-dimensional grid Z^d. We show that the L^1-norm of F_A is considerably larger than log|A| when A is a subset of Z^d with multidimensional structure. We furthermore prove similar lower bounds for sets in Z, which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno & Smith and Konyagin.