{"paper":{"title":"Positive dyadic density for rational weighted binary expansions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Han Wang, Jose Maria Grau ribas","submitted_at":"2026-06-23T11:45:11Z","abstract_excerpt":"Let \\(P/Q\\in\\mathbb Q\\), \\(Q\\ge1\\), and suppose \\[\n  \\sum_{n\\ge1} n d_n2^{-n}=P/Q,\\qquad d_n\\in\\{0,1\\}, \\] has infinite support \\(S=\\{n:d_n=1\\}\\). We prove that \\(S\\) has positive density on all sufficiently large dyadic blocks: there is \\(c_Q>0\\), depending only on \\(Q\\), such that \\[\n  A_S(2X)-A_S(X)\\ge c_QX \\] for every sufficiently large dyadic \\(X\\), where \\(A_S(X)=\\#(S\\cap[1,X])\\). Hence every increasing sequence \\(a_1<a_2<\\cdots\\) with \\(a_n/n\\to\\infty\\) gives an irrational series \\(\\sum_{n\\ge1}a_n2^{-a_n}\\), settling Erd\\H{o}s Problem~260. The proof uses only the integral carry recurre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24972","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/2606.24972/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"}