## 189 Citations

The Typical Structure of Sets With Small Sumset

- MathematicsInternational Mathematics Research Notices
- 2019

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the…

Efficient arithmetic regularity and removal lemmas for induced bipartite patterns

- Mathematicsdiscrete Analysis
- 2019

Efficient arithmetic regularity and removal lemmas for induced bipartite patterns, Discrete Analysis 2019:3, 14 pp.
This paper provides a common extension of two recent lines of work: the study of…

On the Bogolyubov–Ruzsa lemma

- Mathematics
- 2012

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|.

LINEAR FORMS AND QUADRATIC UNIFORMITY FOR FUNCTIONS ON n p

- Mathematics
- 2011

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system…

Set addition in boxes and the Freiman-Bilu theorem

- Mathematics
- 2005

We show that if A is a large subset of a box in Z^d with dimensions L_1 >= L_2 >= ... >= L_d which are all reasonably large, then |A + A| > 2^{d/48}|A|.
By combining this with Chang's quantitative…

On a question of Erdős and Moser

- Mathematics
- 2005

For two finite sets of real numbers A and B, one says that B is sum-free with respect to A if the sum set {b + b | b, b ∈ B, b 6= b} is disjoint from A. Forty years ago, Erdős and Moser posed the…

Planar sets containing no three collinear points and non-averaging sets of integers

- MathematicsDiscret. Math.
- 2002

On Doubling and Volume: Chains

- Mathematics
- 2016

The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$,…

Finding Linear Patterns of Complexity One

- Mathematics
- 2013

We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct…

Freiman's theorem in an arbitrary nilpotent group

- Mathematics
- 2012

We prove a Freiman–Ruzsa‐type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K ‐approximate group A in an s ‐step nilpotent group G is contained in a coset nilprogression…

## References

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What is the structure of K if K+K is small?

- Mathematics
- 1987

I . Let K = {ao,al . . . . . ak_l } be a f i n i t e set of in teger vectors , l e t 2K = K + K = {~ E~m : # = ai + a j l a i ' a j E K} be i t s sum set , and l e t T = 12K I be the c a r d i n a l…

A Tribute to Paul Erdős: On arithmetic progressions in sums of sets of integers

- Mathematics
- 1990

3 Proof of Theorem 1 9 3.1 Estimation of the g1 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Estimation of the g3 term. . . . . . . . . . . . . . . . . . . . . . . .…

Integer Sum Sets Containing Long Arithmetic Progressions

- Mathematics
- 1992

the Schnirelmann and lower asymptotic densities respectively of d. According to Schnirelmann theory (see [9]), if 1 > as/ > 0 and Oes/ then a(2s/) ^ 2a(s/)-a(s/) > a(s/); and if a(s/)^\ then 2s/ =…