{"paper":{"title":"Syndetic submeasures and partitions of $G$-spaces and groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.GR","authors_text":"Igor Protasov, Sergiy Slobodianiuk, Taras Banakh","submitted_at":"2012-10-22T04:58:58Z","abstract_excerpt":"We prove that for every number k each countable infinite group $G$ admits a partition $G=A\\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\\subset G$ the sets $KA$ and $KB$ are not thick. The proof is based on the fact that $G$ possesses a syndetic submeasure, i.e., a left-invariant submeasure $\\mu:\\mathcal P(G)\\to[0,1]$ such that for each $\\epsilon > 1/|G|$ and subset $A\\subset G$ with $\\mu(A)<1$ there is a set $B\\subset G\\setminus A$ such that $\\mu(B)<\\epsilon$ and $FB=G$ for some finite subset $F\\subset G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5804","kind":"arxiv","version":5},"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"}