# Near optimal bounds in Freiman's theorem

@article{Schoen2011NearOB,
title={Near optimal bounds in Freiman's theorem},
author={Tomasz Schoen},
journal={Duke Mathematical Journal},
year={2011},
volume={158},
pages={1-12}
}
• T. Schoen
• Published 15 May 2011
• Mathematics
• Duke Mathematical Journal
We prove that if for a finite set A of integers we have |A+A| ≤ K|A|, then A is contained in a generalized arithmetic progression of dimension at most K −1/2 and size at most exp(K −1/2 )|A| for some absolute constant C. We also discuss a number of applications of this result.

### 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|.

### Additive Volume of Sets Contained in Few Arithmetic Progressions

• Mathematics
Integers
• 2019
The formula for the largest volume of a finite set of integers with given cardinality and doubling is extended to sets of every dimension and proved for sets composed of three segments, giving structural results for the extremal case.

### Small Doubling in Groups

• Mathematics
• 2013
Let A be a subset of a group G = (G; ·). We will survey the theory of sets A with the property that |A · A|≤K|A|, where A · A = {a1a2: a1; a2 ∈ A}. The case G = (ℤ; +) is the famous Freiman-Ruzsa

### Some new inequalities in additive combinatorics

In the paper we find new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group. We apply the inequalities to obtain new upper bounds for the

### Sums of Linear Transformations in Higher Dimensions

In this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 10 depending only on q, s and d such that for any finite subset A of ℝd that is

### ADDITIVE DIMENSION AND A THEOREM OF SANDERS

• Mathematics
Journal of the Australian Mathematical Society
• 2015
We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets of an abelian group.

### Roth’s theorem in many variables

• Mathematics
• 2011
We prove that if A ⊆ {1, ..., N} has no nontrivial solution to the equation x1 + x2 + x3 + x4 + x5 = 5y, then $|A| \ll Ne^{ - c(\log N)^{1/7} }$, c > 0. In view of the well-known Behrend

### On additive shifts of multiplicative subgroups

• Mathematics
• 2012
It is proved that for an arbitrary subgroup and any distinct nonzero elements we have under the condition that , where , are some sequences of positive numbers such that as . Furthermore, it is shown

### An elementary additive doubling inequality

We prove an elementary additive combinatorics inequality, which says that if $A$ is a subset of an Abelian group, which has, in some strong sense, large doubling, then the difference set A-A has a