Finite products sets and minimally almost periodic groups (Q5963423)

\textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] proved a conjecture of \textit{R. L. Graham} and \textit{B. L. Rothschild} [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] by showing that in any finite partition of the natural numbers one of the atoms contains an IP (or finite sum) set. Erdős asked if this important result could have a density form, just as van der Waerden's theorem (that one of the atoms contains arbitrarily long arithmetic progressions) has a density version in Szemerédi's theorem (that a subset of the natural numbers with positive upper density must contain arbitrarily long arithmetic progressions), specifically asking if a set of positive upper density contains a translate of a finite sum set. Straus [unpublished] showed no such density version could hold by constructing arbitrarily large density subsets of the natural numbers with the property that no translate contains a finite sum set.NEWLINENEWLINEHere Straus' result is explored for (locally compact, second countable) amenable groups in place of the natural numbers. The main results (which are too technical to state here) give a characterization of those locally compact second countable amenable groups in which a density version of Hindman's theorem holds, and those countable amenable groups in which a two-sided density version of Hindman's theorem holds. The failure to find such a result is in both cases related to having many finite-dimensional unitary representations, while the success is related to the ergodic property of weak-mixing via an appropriate new form of the Furstenberg correspondence principle.