Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sumsets of measurable sets


Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 78 (1980), 59-63
MSC: Primary 10L05; Secondary 28A05
DOI: https://doi.org/10.1090/S0002-9939-1980-0548085-3
MathSciNet review: 548085
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathcal{A}_1},{\mathcal{A}_2}, \ldots ,{\mathcal{A}_n}$ be Lebesgue measurable sets of positive real numbers such that $ {\inf \mathcal{A}_i} = 0$ for all i. Let $ \mu $ denote Lebesgue measure and let $ {\mu _ \ast }$ denote inner Lebesgue measure. If $ \sum \nolimits_{i = 1}^n \mu ({\mathcal{A}_i} \cap [0,t]) \geqslant \gamma t$ for some $ \gamma \leqslant 1$ and all $ t \leqslant x$, then

$\displaystyle {\mu _ \ast }(({\mathcal{A}_1} + {\mathcal{A}_2} + \cdots + {\mathcal{A}_n}) \cap [0,x]) \geqslant \gamma x.$

This generalizes results of Dyson and Macbeath.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10L05, 28A05

Retrieve articles in all journals with MSC: 10L05, 28A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0548085-3
Keywords: Sums of sets of real numbers, sums of sequences, additive number theory, addition theorems
Article copyright: © Copyright 1980 American Mathematical Society