Duke Mathematical Journal

A polynomial bound in Freiman's theorem

Mei-Chu Chang

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Abstract

In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1).

Let A be a finite set such that | A+A |<α| A | . Then A is contained in a proper d-dimensional progression P, where d[ α1 ] and log( | P |/| A | )<C α 2 ( logα ) 3 .

Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.

Article information

Source
Duke Math. J. Volume 113, Number 3 (2002), 399-419.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575313

Digital Object Identifier
doi:10.1215/S0012-7094-02-11331-3

Mathematical Reviews number (MathSciNet)
MR1909605

Zentralblatt MATH identifier
1035.11048

Subjects
Primary: 11P70: Inverse problems of additive number theory, including sumsets
Secondary: 11B13: Additive bases, including sumsets [See also 05B10] 11B25: Arithmetic progressions [See also 11N13]

Citation

Chang, Mei-Chu. A polynomial bound in Freiman's theorem. Duke Math. J. 113 (2002), no. 3, 399--419. doi:10.1215/S0012-7094-02-11331-3. https://projecteuclid.org/euclid.dmj/1087575313


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