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Zbl 1247.11130
Serra, Oriol; Zémor, Gilles
Large sets with small doubling modulo $p$ are well covered by an arithmetic progression. (English)
[J] Ann. Inst. Fourier 59, No. 5, 2043-2060 (2009). ISSN 0373-0956; ISSN 1777-5310/e

Summary: We prove that there is a small but fixed positive integer $\varepsilon$ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $|2S|\le (2+\varepsilon)|S|$ and $2(|2S|)-2|S|+3\le p$ is contained in an arithmetic progression of length $|2S|-|S|+1$. This is the first result of this nature which places no unnecessary restrictions on the size of $S$.

MSC 2000:: *11P70 Inverse problems of additive number theory
11B30
11B25 Arithmetic progressions

Keywords: sumset; arithmetic progression; additive combinatorics

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