{"paper":{"title":"Solution to a conjecture of Alon, D\\k{e}bski, Grytczuk and Przyby\\l{}o on fixed-cardinality arithmetic progressions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gang Yang, Meiqin Wei, Yaping Mao, Zhao Wang","submitted_at":"2026-07-07T10:22:54Z","abstract_excerpt":"Fix a positive integer $n$, and put $B_d=\\{d,2d,\\ldots,nd\\}$. Let $M_k(n)$ be the least integer $m$ for which one translate of each of $B_1,\\ldots,B_k$ can be placed pairwise disjointly in $[m]$. We prove that, for every $\\eps\\in(0,1)$ and all sufficiently large $k$, one has $M_k(n)\\le n\\lceil(1+\\eps)k\\rceil$. Since the trivial counting bound gives $M_k(n)\\ge nk$, it follows that $M_k(n)=(1+o(1))nk$ for every fixed $n$. This confirms a conjecture of Alon, D\\k{e}bski, Grytczuk and Przyby\\l{}o on prescribed-difference packings of fixed-cardinality arithmetic progressions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06113","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/2607.06113/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"}