Additive Combinatorics and its Applications in Theoretical Computer Science

Shachar Lovett
2017 Theory of Computing  
Additive combinatorics (or perhaps more accurately, arithmetic combinatorics) is a branch of mathematics which lies at the intersection of combinatorics, number theory, Fourier analysis and ergodic theory. It studies approximate notions of various algebraic structures, such as vector spaces or fields. In recent years, several connections between additive combinatorics and theoretical computer science have been discovered. Techniques and results from additive combinatorics have been applied to
more » ... oblems in coding theory, property testing, hardness of approximation, computational complexity, communication complexity, randomness extraction and pseudo-randomness. The goal of this survey is to provide an introduction to additive combinatorics for students and researchers in theoretical computer science, to illustrate some of the exciting connections to classical problems in theoretical computer science, and to describe the many open problems that remain. • The book "Additive Combinatorics" by Tao and Vu [73] gives a detailed description of many results in additive combinatorics and their applications, mainly in number theory. • A mini-course on additive combinatorics by Barak et al. [6] explores basic ideas in additive combinatorics and some of their applications in computer science. • The survey "Finite field models in additive combinatorics" by Green [35] . • The survey "Selected Results in Additive Combinatorics: An Exposition" by Viola [75] covers the basic results related to set addition and their application in Samorodnitsky's theorem [61] on linearity testing for functions with multiple outputs. • The survey "The Polynomial Freiman-Ruzsa conjecture" by Green [36] explores Katalin Marton's "Polynomial Freiman-Ruzsa conjecture," one of the central open problems in this area. A more specialized survey by the author [49] gives an exposition of a recent important result by Sanders [65] which gets close to proving this conjecture. • The survey "Additive combinatorics and theoretical computer science" by Trevisan [74] covers in high level many of the recent advances in additive combinatorics, and discusses their relations to problems arising in theoretical computer science. • The survey "Additive combinatorics: with a view towards computer science and cryptography-an exposition" by Bibak [13] covers a number of results in additive combinatorics, motivated by cryptographic applications. • The survey "Additive combinatorics over finite fields: New results and applications" by Shparlinski [66] discusses recent results in additive combinatorics. Acknowledgements. I thank the anonymous referees for a careful reading of this manuscript. Set addition Let G be an Abelian group, where a motivating example to have in mind is G = F n (i. e., a vector space over a field). Let A be a finite subset of the group. The sumset of A is defined to be the set It is simple to see that |2A| ≥ |A|. Equality holds only if A is either empty, or a subgroup of G, or a coset of a subgroup. Indeed, if A = / 0 then we can assume 0 ∈ A by shifting A (that is, replacing A by A − a 0 = {a − a 0 : a ∈ A} for some a 0 ∈ A). This does not change the size of A or 2A. Now, since 0 ∈ A we have A ⊆ 2A. Moreover, since by assumption |2A| = |A| then in fact 2A = A. That is, A is a nonempty finite subset closed under addition, and hence is a subgroup of G. The focus of this chapter is on relaxations of this simple claim. We will consider various notions of approximate subgroups and how they relate to each other. Ruzsa calculus Ruzsa calculus is a set of basic inequalities between sizes of sets and their sumsets. Despite being basic, it is very useful. Let A, B be two subsets of an Abelian group G. Define A + B = {a + b : a ∈ A, b ∈ B} and A − B = {a − b : a ∈ A, b ∈ B} to be their sumset and difference set, respectively. We start with the Ruzsa triangle inequality, which is both simple and useful.
doi:10.4086/ dblp:journals/toc/Lovett17 fatcat:z5d45zpq3vgpzk7tshdszrdwni