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

This article is cited in 8 scientific papers (total in 8 papers)

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
Received: 01.03.2007
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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    Citing articles on Google Scholar: Russian citations, English citations
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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  crossref  mathscinet  zmath  isi  elib  scopus
    2. Shkredov I.D., Yekhanin S., “Sets with large additive energy and symmetric sets”, J. Combin. Theory Ser. A, 118:3 (2011), 1086–1093  crossref  mathscinet  zmath  isi  elib  scopus
    3. Sanders T., “On certain other sets of integers”, J. Anal. Math., 116 (2012), 53–82  crossref  mathscinet  zmath  isi  elib  scopus
    4. I. D. Shkredov, “Some new results on higher energies”, Trans. Moscow Math. Soc., 74 (2013), 31–63  mathnet  crossref  mathscinet  zmath  elib
    5. Ilya Sh., “Energies and Structure of Additive Sets”, Electron. J. Comb., 21:3 (2014), P3.44  zmath  isi
    6. A. A. Uvakin, “On Two-Dimensional Sums and Differences”, Math. Notes, 98:4 (2015), 636–652  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Schoen T., Shkredov I.D., “Additive Dimension and a Theorem of Sanders”, J. Aust. Math. Soc., 100:1 (2016), 124–144  crossref  mathscinet  zmath  isi  scopus
    8. A. A. Uvakin, “On Two-Dimensional Sums in Abelian Groups”, Math. Notes, 103:2 (2018), 271–289  mathnet  crossref  crossref  mathscinet  isi  elib
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