{"paper":{"title":"Prime-Power Rarefaction and a Density-One Lower Bound for Erd\\H{o}s Problem 400","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-22T17:47:25Z","abstract_excerpt":"For fixed $k\\ge 2$, let $g_k(n)$ be the greatest excess $a_1+\\cdots+a_k-n$ among positive integers $a_i$ satisfying $a_1!\\cdots a_k!\\mid n!$. We prove that, for every $\\varepsilon>0$, all but $o(x)$ integers $n\\le x$ satisfy \\[ g_k(n)\\ge \\left(\\frac{3(k-1)}{\\log 12}-\\varepsilon\\right)\\log n. \\] We also prove, as $n\\to\\infty$, the pointwise upper bound \\[ g_k(n)\\le (k-1)\\log_2 n+\\log_2\\log n+O_k(1). \\] The central analytic input is uniform phase separation for one or two frequencies on fixed-prime $S$-unit progressions, deduced directly from the finite exceptional-subspace alternative of Drmota"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23661","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/2606.23661/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"}