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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Plünnecke's Theorem for asymptotic densities


Author: Renling Jin
Journal: Trans. Amer. Math. Soc. 363 (2011), 5059-5070
MSC (2010): Primary 11B05, 11B13, 11U10, 03H15
DOI: https://doi.org/10.1090/S0002-9947-2011-05533-9
Published electronically: May 24, 2011
MathSciNet review: 2813407
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Abstract: Plünnecke proved that if $ B\subseteq\mathbb{N}$ is a basis of order $ h>1$, i.e., $ \sigma(hB)=1$, then $ \sigma(A+B)\geqslant\sigma(A)^{1-\frac{1}{h}}$, where $ A$ is an arbitrary subset of $ \mathbb{N}$ and $ \sigma$ represents Shnirel'man density. In this paper we explore whether $ \sigma$ can be replaced by other asymptotic densities. We show that Plünnecke's inequality above is true if $ \sigma$ is replaced by lower asymptotic density $ \underline{d}$ or by upper Banach density $ BD$ but not by upper asymptotic density $ \overline{d}$. The result about $ \underline{d}$ has some interesting consequences such as the inequality $ \underline{d}(A+P)\geqslant\underline{d}(A)^{2/3}$ for any $ A\subseteq\mathbb{N}$, where $ P$ is the set of all prime numbers, and the inequality $ \underline{d}(A+C)\geqslant\underline{d}(A)^{3/4}$ for any $ A\subseteq\mathbb{N}$, where $ C$ is the set of all cubes of nonnegative integers. The result about $ BD$ generalizes Theorem 3 of a 2001 work of the author by reducing the requirement of $ B$ being a piecewise basis to the requirement of $ B$ being an upper Banach basis.


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Additional Information

Renling Jin
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
Email: jinr@cofc.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05533-9
Keywords: Plünnecke’s inequality, Shnirel’man density, lower asymptotic density, upper asymptotic density, upper Banach density, basis, nonstandard analysis
Received by editor(s): February 9, 2009
Published electronically: May 24, 2011
Additional Notes: This research was supported in part by NSF RUI grant DMS–#0500671. The author would also like to express his gratitude to Professor Georges Grekos, Professor François Hennecart, and other mathematicians in Laboratoire des Mathématiques Unifiées de Saint-Etienne (LaMUSE) for their hospitality and stimulating discussions during the author’s visit there in the summer of 2007 when the essential part of this research project was developed.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.