{"paper":{"title":"A first-exit proof of Cusick's sum-of-digits conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kaimin Cheng","submitted_at":"2026-06-22T14:21:11Z","abstract_excerpt":"We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer \\(t\\ge 1\\) we show that \\[ c_t:=\\lim_{N\\to\\infty}\\frac{1}{N} \\#\\{0\\le n<N:\\ s_2(n+t)\\ge s_2(n)\\}>\\frac{1}{2}, \\] and in fact obtain the explicit bound \\[ c_t\\ge \\frac{1}{2}+2^{-2s_2(t)-1}, \\] where \\(s_2(m)\\) denotes the number of ones in the binary expansion of \\(m\\). The proof is based on an exact deconvolution which replaces the distribution of \\(s_2(n+t)-s_2(n)\\) by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23398","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/2606.23398/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"}