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Erschienen in:
2014 | OriginalPaper | Buchkapitel
Density Versions of Plünnecke Inequality: Epsilon-Delta Approach
verfasst von : Renling Jin
Erschienen in: Combinatorial and Additive Number Theory
Verlag: Springer New York
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Abstract
We discuss whether Plünnecke’s inequality for Shnirel’man density with respect to Shnirel’man basis can be generalized to other densities with respect to other concepts of basis. We show behavioral disparities between lower densities and upper densities on Plünnecke’s inequality. We provide standard proofs for the generalizations of Plünnecke’s inequality to lower asymptotic density and upper Banach density. These two results were proved before by the author using nonstandard analysis. A similar generalization to upper asymptotic density is not true. We also provide a standard proof for a new generalization to lower Banach density with respect to upper Banach basis. In the last section we present a simplified proof of Plünnecke’s inequality for Shnirel’man density with respect to Shnirel’man basis without introducing the impact function.