Sumset size races for measurable sets
Melvyn B. Nathanson

TL;DR
This paper demonstrates the flexibility in the sizes of sumsets of measurable sets within locally compact abelian groups, showing that any prescribed order or differences can be realized through appropriate set constructions.
Contribution
It establishes that for any given orderings and differences, there exist measurable sets in groups and the real line that realize these configurations in their sumsets.
Findings
Constructs measurable sets with prescribed sumset size orderings.
Shows existence of sets with specified sumset size differences.
Extends sumset size control to general locally compact abelian groups.
Abstract
Let be a locally compact abelian group with Haar measure . For integers and and for any -tuples , there exist measurable subsets of such that the -tuple has the same relative order as the -tuple for all . For integers for and , there are Lebesgue measurable sets in such that for all and .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Mathematical Dynamics and Fractals
