Kneser-type theorems for countable amenable groups
Michael Björklund, Alexander Fish
We develop in this paper a general ergodic-theoretical technique which reduces to a large extent the study of lower bounds on asymptotic densities (with respect to some fixed Følner sequence) of product sets in a countable amenable group G, to the study of lower bounds of products of Borel sets in a metrizable compactification of G, for which a plethora of classical (as well as more recent) results are known. We apply this technique to extend Kneser’s classical sum set bound in the additive group (Z, +) (a density analogue of Mann’s celebrated theorem) to asymptotic densities on general countable amenable groups, as well as to answer some old questions concerning the sharpness of this bound. Roughly speaking, our main combinatorial results assert that if the product of two "large" sets in G is "small", then the "left" addend is very well controlled by either a proper periodic set or a Sturmian set in G. As an application of our results, we provide a classification of "spread-out" approximative subgroups of spread two in a general countable amenable group. We also develop a "Counterexample Machine" which produces in certain two-step solvable groups, pairs of large subsets with "small" product sets whose "right" (but not left) addend fails to be "nicely" controlled by well-structured Bohr sets.
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