{"paper":{"title":"A Local Classification of Four-Element Multiple Sumsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Minkyu Jung","submitted_at":"2026-07-21T04:25:11Z","abstract_excerpt":"For a finite set $A\\subset\\mathbb{Z}$, write $hA$ for its $h$-fold sumset, and let \\[ R(h,k)=\\{|hA|:A\\subset\\mathbb{Z},\\ |A|=k\\}. \\] We determine the part of $R(h,4)$ lying between $4h+2$ and $6h-4$: for $h=4$ the only value is $5h-1$, while for $h\\geq 5$ the only values are $5h-1$ and $5h+1$. This proves Rajagopal's conjectured gap $5h\\notin R(h,4)$ for every $h\\geq 4$. For $h\\geq 6$, it also yields the new missing interval $[5h+2,6h-4]$, which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18694","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.18694/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}