# Generalized arithmetical progressions and sumsets

```@article{Ruzsa1994GeneralizedAP,
title={Generalized arithmetical progressions and sumsets},
author={Imre Z. Ruzsa},
journal={Acta Mathematica Hungarica},
year={1994},
volume={65},
pages={379-388}
}```
• I. Ruzsa
• Published 1 December 1994
• Mathematics
• Acta Mathematica Hungarica
189 Citations
The Typical Structure of Sets With Small Sumset
• Mathematics
International 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
• Mathematics
discrete 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
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
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
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
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
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?
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
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/ =