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 Mat. Zametki, 2008, Volume 84, Issue 6, Pages 927–947 (Mi mz3996)

On Sets with Small Doubling Property

I. D. Shkredov

Lviv Polytechnic National University

Abstract: Suppose that $G$ is an arbitrary Abelian group and $A$ is any finite subset $G$. A set $A$ is called a set with small sumset if, for some number $K$, we have $|A+A|\le K|A|$. The structural properties of such sets were studied in the papers of Freiman, Bilu, Ruzsa, Chang, Green, and Tao. In the present paper, we prove that, under certain constraints on $K$, for any set with small sumset, there exists a set $\Lambda$, $\Lambda\ll_{\varepsilon}K\log|A|$, such that $|A\cap \Lambda|\gg |A|/K^{1/2+\varepsilon}$, where $\varepsilon>0$. In contrast to the results of the previous authors, our theorem is nontrivial even for a sufficiently large $K$. For example, for $K$ we can take $|A|^\eta$, where $\eta>0$. The method of proof used by us is quite elementary.

Keywords: Abelian group, sumset (Minkowski sum), set with small doubling property, arithmetic progression, connected set, dissociate set, Cauchy–Bunyakovskii inequality

DOI: https://doi.org/10.4213/mzm3996

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English version:
Mathematical Notes, 2008, 84:6, 859–878

Bibliographic databases:

UDC: 512.54
Revised: 02.04.2008

Citation: I. D. Shkredov, “On Sets with Small Doubling Property”, Mat. Zametki, 84:6 (2008), 927–947; Math. Notes, 84:6 (2008), 859–878

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz/v84/i6/p927

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This publication is cited in the following articles:
1. Sanders T., “A quantitative version of the non-abelian idempotent theorem”, Geom. Funct. Anal., 21:1 (2011), 141–221
2. Shkredov I.D., Yekhanin S., “Sets with large additive energy and symmetric sets”, J. Combin. Theory Ser. A, 118:3 (2011), 1086–1093
3. Sanders T., “On certain other sets of integers”, J. Anal. Math., 116 (2012), 53–82
4. I. D. Shkredov, “Some new results on higher energies”, Trans. Moscow Math. Soc., 74 (2013), 31–63
5. Ilya Sh., “Energies and Structure of Additive Sets”, Electron. J. Comb., 21:3 (2014), P3.44
6. A. A. Uvakin, “On Two-Dimensional Sums and Differences”, Math. Notes, 98:4 (2015), 636–652
7. Schoen T., Shkredov I.D., “Additive Dimension and a Theorem of Sanders”, J. Aust. Math. Soc., 100:1 (2016), 124–144
8. A. A. Uvakin, “On Two-Dimensional Sums in Abelian Groups”, Math. Notes, 103:2 (2018), 271–289
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