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The Typical Structure of Sets with Small Sumset

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed \(\lambda > 2\) and every \(k \geqslant (\log n)^4\): if \(\omega \to...

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Bibliographic Details
Published in:arXiv.org 2020-10
Main Authors: Campos, Marcelo, Collares, MaurĂ­cio, Morris, Robert, Morrison, Natasha, Souza, Victor
Format: Article
Language:English
Online Access:Get full text
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Summary:In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed \(\lambda > 2\) and every \(k \geqslant (\log n)^4\): if \(\omega \to \infty\) as \(n \to \infty\) (arbitrarily slowly), then almost all sets \(A \subset [n]\) with \(|A| = k\) and \(|A + A| \leqslant \lambda k\) are contained in an arithmetic progression of length \(\lambda k/2 + \omega\).
ISSN:2331-8422