Small product sets in compact groups

2014/02/07 by Michael Björklund, Björklund, Michael
Mathematics · #11P70 #22C05 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT #msc:11P70 #msc:22C05

paper · pdf · doi:10.48550/arxiv.1402.1618

24 pages, no figures. A re-write of an earlier version - significantly shorter, and (hopefully) better structured. Comments are welcome! Accepted in Fundamenta Math

arxiv created 2016/12/04 · arxiv updated 2016/12/06

Abstract

We show in this paper that a sub-critical pair (A,B) of sufficiently "spread-out" Borel sets in a compact and second countable group K with an abelian identity component, must reduce to a Sturmian pair in either \bT or \bT \rtimes \-1,1\. This extends a classical result of Kneser.

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• The Structure of Critical Product Sets
• Critical pairs in abelian groups and Kemperman's structure theorem
• Self-small products of abelian groups
• An inverse theorem: when the measure of the sumset is the sum of the measures in a locally compact abelian group
• A Study in Locally Compact Groups