## Duke Mathematical Journal

### Near optimal bounds in Freiman's theorem

Tomasz Schoen

#### Abstract

We prove that if for a finite set $A$ of integers we have $|A+A|\le K|A|$, then $A$ is contained in a generalized arithmetic progression of dimension at most $K^{1+C(\log K)^{-1/2}}$ and of size at most ${\rm exp}({K^{1+C(\log K)^{-1/2}}})|A|$ for some absolute constant $C$. We also discuss a number of applications of this result.

#### Article information

Source
Duke Math. J. Volume 158, Number 1 (2011), 1-12.

Dates
First available in Project Euclid: 3 May 2011

http://projecteuclid.org/euclid.dmj/1304429491

Digital Object Identifier
doi:10.1215/00127094-1276283

Mathematical Reviews number (MathSciNet)
MR2794366

Zentralblatt MATH identifier
05904309

#### Citation

Schoen, Tomasz. Near optimal bounds in Freiman's theorem. Duke Math. J. 158 (2011), no. 1, 1--12. doi:10.1215/00127094-1276283. http://projecteuclid.org/euclid.dmj/1304429491.

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