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Published in:
2006 | OriginalPaper | Chapter
Piecewise-Bohr Sets of Integers and Combinatorial Number Theory
Authors : Vitaly Bergelson, Hillel Furstenberg, Benjamin Weiss
Published in: Topics in Discrete Mathematics
Publisher: Springer Berlin Heidelberg
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We use ergodic-theoretical tools to study various notions of “large” sets of integers which naturally arise in theory of almost periodic functions, combinatorial number theory, and dynamics. Call a subset of N a Bohr set if it corresponds to an open subset in the Bohr compactification, and a piecewise Bohr set (PWB) if it contains arbitrarily large intervals of a fixed Bohr set. For example, we link the notion of PWB-sets to sets of the form A+B, where A and B are sets of integers having positive upper Banach density and obtain the following sharpening of a recent result of Renling Jin.
Theorem. If A and B are sets of integers having positive upper Banach density, the sum set A+B is PWB-set.