{"paper":{"title":"The structure of sets with cube-avoiding sumsets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Peter Keevash, Thomas Karam","submitted_at":"2024-11-21T14:07:46Z","abstract_excerpt":"We prove that if $d \\ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\\dots+E_d$ avoids $Z_0^n$ then $E_1, \\dots, E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1' \\times G^{I^c}, \\dots, E_d' \\times G^{I^c}$, where the size of $I \\subset [n]$ is non-zero and independent of $n$, and $E_1',\\dots,E_d'$ are subsets of $G^{I}$ such that the sumset $E_1'+\\dots+E_d'$ avoids $Z_0^I$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14145","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/2411.14145/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"}