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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.
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
Linking options:
https://www.mathnet.ru/eng/mzm3996https://doi.org/10.4213/mzm3996 https://www.mathnet.ru/eng/mzm/v84/i6/p927
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