Plünnecke’s Theorem for asymptotic densities
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- by Renling Jin PDF
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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.References
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Additional Information
- Renling Jin
- Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
- Email: jinr@cofc.edu
- 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.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2813407