On the Bogolyubov-Ruzsa lemma

Abstract

Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|.

28 pp. Corrected typos. Added appendix on model setting

References in corpus (4)

• An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm
• Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak
• On certain other sets of integers
• Linear forms and quadratic uniformity for functions on ${\mathbb{Z}}_N$

Cited by in corpus (34)

• Energies and structure of additive sets
• Open Problems in Analysis of Boolean Functions
• Arithmetic progressions in sumsets and L^p-almost-periodicity
• Query complexity and the polynomial Freiman-Ruzsa conjecture
• Polynomial bound for the partition rank vs the analytic rank of tensors
• Additive dimension and a theorem of Sanders
• Sumsets and entropy revisited
• Approximate algebraic structure
• The structure of approximate groups
• An inverse theorem for Freiman multi-homomorphisms
• The Kelley--Meka bounds for sets free of three-term arithmetic progressions
• A quantitative inverse theorem for the $U^4$ norm over finite fields
• Boolean functions with small spectral norm, revisited
• Large values of the additive energy in R^d and Z^d
• Arithmetic progressions in sets of small doubling
• Recent advances on the log-rank conjecture in communication complexity
• Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle
• An introduction to higher energies and sumsets
• Approximate subgroups with bounded VC-dimension
• Sparse graph counting and Kelley-Meka bounds for binary systems
• Growth in groups: ideas and perspectives
• A note on inverse results of random walks in Abelian groups
• Additive combinatorics with a view towards computer science and cryptography: An exposition
• Some properties of lower level-sets of convolutions
• Order-preserving Freiman isomorphisms
• A bound on the multiplicative energy of a sum set and extremal sum-product problems
• Freiman's theorem in an arbitrary nilpotent group
• The PFR Conjecture Holds for Two Opposing Special Cases
• Revisiting the Sanders-Freiman-Ruzsa Theorem in ${\mathbb{F}}_p^n$ and its Application to Non-malleable Codes
• Approximating Sumset Size
• Approximate Subloops in Moufang Loops
• On the size of $A+\lambda A$ for algebraic $\lambda$
• A Bilinear Bogolyubov Argument in Abelian Groups
• Polynomials and Primes in Generalized Arithmetic Progressions (Revised Version)