{"paper":{"title":"On Asymptotic Approximate Groups of Integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bela Bajnok","submitted_at":"2016-03-21T19:37:20Z","abstract_excerpt":"Let $r$ be a positive integer, and let $A$ be a nonempty finite set of at least two integers. We let $\\tilde{C}_r(A)$ denote the {\\em asymptotic $r$-covering number} of $A$, that is, the smallest integer value of $l$ for which, for all sufficiently large positive integers $h$, the $rh$-fold sumset of $A$ is contained in at most $l$ translates of the $h$-fold sumset of $A$. Nathanson proved that $\\tilde{C}_r(A)$ is always at most $r+1$; here we extend this result to prove that $\\tilde{C}_r(A)$ is always at least $r$, and determine all sets $A$ for which $\\tilde{C}_r(A)=r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06553","kind":"arxiv","version":1},"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"}