Abstract

The polynomial Freiman-Ruzsa conjecture is one of the most important conjec- tures in additive combinatorics. It asserts that one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying pa- rameters. This conjecture has also found several applications in theoretical computer science. Recently, Tom Sanders proved a weaker version of the conjecture, with a quasi-polynomial loss in parameters. The aim of this note is to make his proof accessible to the theoretical computer science community, and in particular to readers who are less familiar with additive combinatorics.

Highlights

  • Let A be a finite subset of an abelian group G

  • We note that many of the results discussed here can be extended to vector spaces over larger fields, general abelian groups, and sometimes even to non-abelian groups

  • The aim of this note is to give a detailed exposition of the following breakthrough result of Sanders [22], who proved a weaker version of the Freiman-Ruzsa conjecture with a quasi-polynomial loss in parameters

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Summary

Introduction

Let A be a finite subset of an abelian group G. The aim of this note is to give a detailed exposition of the following breakthrough result of Sanders [22], who proved a weaker version of the Freiman-Ruzsa conjecture with a quasi-polynomial loss in parameters. As noted before, his result extends to more general abelian groups, but we focus on Fn2 for simplicity of exposition. We discuss briefly the how the polynomial Freiman-Ruzsa conjecture (Conjecture 1.2) and Sanders’ result (Theorem 1.4) can be extended to general abelian groups. These include the book “Additive Combinatorics” by Tao and Vu [23]; a mini-course on additive combinatorics by Barak et al [2]; a survey covering selected results in additive combinatorics by Viola [25]; a survey on additive combinatorics and theoretical computer science by Trevisan [24]; a survey with a focus on applications in cryptography by Bibak [5]; and a survey on additive combinatorics and its applications in computer science by the author [15]

Proof overview
Preliminaries
Reductions
Reduction 1
Reduction 2
Existence of a large near-invariant set
A Fourier-analytic argument

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