{"paper":{"title":"Nonstandard Analysis and the sumset phenomenon in arbitrary amenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Martino Lupini, Mauro Di Nasso","submitted_at":"2012-11-18T11:13:25Z","abstract_excerpt":"Beiglboeck, Bergelson and Fish proved that if subsets A,B of a countable discrete amenable group G have positive Banach densities a and b respectively, then the product set AB is piecewise syndetic, i.e. there exists k such that the union of k-many left translates of AB is thick. Using nonstandard analysis we give a shorter alternative proof of this result that does not require G to be countable, and moreover yields the explicit bound that k is not greater than 1/ab. We also prove with similar methods that if $\\{A_i\\}$ are finitely many subsets of G having positive Banach densities $a_i$ and G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4208","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}