There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over R or C, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective pl...

Abstract: Inspired by the ”rough classification” ideas from additive combinatorics, Soundararajan and I have recently introduced the notion of pretentiousness into analytic number theory. Besides giving a more accessible description of the ideas behind the proofs of several wellknown difficult results of analytic number theory, it has allowed us to strengthen several results, like the PolyaVino...

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL2(Z/pZ) as the basic example, as well as permutation groups. The emphasis will lie on the ideas behind the methods.

We consider the problem of recovering a hidden element s of a finite field Fq of q elements from queries to an oracle that for a given x ∈ Fq returns (x+ s) for a given divisor e | q− 1. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to ...

Turán problems in extremal combinatorics concern asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful semidefinite programming based methods to find sums of squares that establish edge density inequalities in Turán problems. Working with polynomial analogs of the flag algebra entities...

Chang’s lemma is a useful tool in additive combinatorics and the analysis of Boolean functions. Here we give an elementary proof using entropy. The constant we obtain is tight, and we give a slight improvement in the case where the variables are highly biased. 1 The lemma For S ∈ {0, 1}, let χk : {±1} n → R denote the character χS(x) = ∏ i∈S xi . For any function f : {±1} → R, we can then defin...

1. Introduction. Let X be an n-element set and F C 2 X a family of distinct subsets of X. Suppose that the members of F satisfy some conditions. What is the maximum (or minimum) value of |F|—this is the generic problem in extremal set theory. There have been far too many papers and results in this area to be overviewed in such a short paper. Therefore, we will only deal with some intersection t...