The Typical Structure of Sets With Small Sumset

Author:

Campos Marcelo1,Collares Maurício2,Morris Robert1,Morrison Natasha3,Souza Victor1

Affiliation:

1. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, 22460-320, Brazil

2. Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil

3. Department of Mathematics and Statistics, University of Victoria, David Turpin Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

Abstract

Abstract 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 \rightarrow \infty $ as $n \rightarrow \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 $.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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2. A refinement of the Cameron–Erd̋s conjecture;Alon;Proc. London Math. Soc. (3),2014

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4. Sharp bound on the number of maximal sum-free subsets of integers;Balogh;J. Eur. Math. Soc.,2018

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Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. The Typical Approximate Structure of Sets with Bounded Sumset;SIAM Journal on Discrete Mathematics;2023-07-10

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