{"paper":{"title":"The martingale evolution of probability measures defined via the sum-of-digits functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Cusick conjecture on sum-of-digits densities follows from a general median-preserving property of martingales on binary trees.","cross_cats":["math.NT"],"primary_cat":"math.PR","authors_text":"Dawid Tar{\\l}owski","submitted_at":"2026-05-09T02:41:38Z","abstract_excerpt":"Let $s(n)$ denote the number of ones in the binary expansion of a natural number $n\\in\\mathbb{N}$. For any $t\\in\\mathbb{N}$ and $d\\in\\mathbb{Z}$, let $\\mu_t(d)$ denote the asymptotic density of the set of those natural numbers $n$ for which $s(n+t)-s(n)=d$. It is well known that $\\mu_t$ are properly defined probability measures on $\\mathbb{Z}$, and the Cusick conjecture states that $\\mu_t(\\mathbb{N})>\\frac{1}{2}$ for any $t\\in\\mathbb{N}$. In this paper, we investigate the properties of the family $\\{\\mu_t\\}_{t\\in\\mathbb{N}}$ by reindexing the odd integers via a suitable partial order. This con"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the Cusick conjecture is a special case of a more general claim about the asymmetric evolution of the binary trees associated to the martingale","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"We will assume that the random walk starts from zero, and thus we will work with the family of measures P_t determined by the convolution μ_t=μ_1∗P_t. The reindexing of odd integers via partial order leads to the nonautonomous dynamics on pairs of measures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A martingale from binary tree stopping times describes the evolution of sum-of-digits difference measures, generalizing the Cusick conjecture to asymmetric tree growth.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Cusick conjecture on sum-of-digits densities follows from a general median-preserving property of martingales on binary trees.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"cd63f0b7e96352a1fca6b10869586f89775df641d77b3a0dd1b371c35277c27d"},"source":{"id":"2605.08624","kind":"arxiv","version":2},"verdict":{"id":"ac8eeb90-8f40-401d-a59e-468111f9982a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T00:57:24.774810Z","strongest_claim":"the Cusick conjecture is a special case of a more general claim about the asymmetric evolution of the binary trees associated to the martingale","one_line_summary":"A martingale from binary tree stopping times describes the evolution of sum-of-digits difference measures, generalizing the Cusick conjecture to asymmetric tree growth.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"We will assume that the random walk starts from zero, and thus we will work with the family of measures P_t determined by the convolution μ_t=μ_1∗P_t. The reindexing of odd integers via partial order leads to the nonautonomous dynamics on pairs of measures.","pith_extraction_headline":"The Cusick conjecture on sum-of-digits densities follows from a general median-preserving property of martingales on binary trees."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.08624/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T09:02:02.035615Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:37:06.968097Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T14:31:18.112673Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:58:51.330170Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"4f5dd2aa9550c7ea650f0a8ac965c64438d0172f41622d6a2f10a877fe19e10f"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"213a5d64484edac3bfb04f8b276e9b209b95bc5479a8c5ecd551ac909a1b5f42"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}