{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:I5VDY4DKZVLKCJEWPZADKBR6ZZ","short_pith_number":"pith:I5VDY4DK","schema_version":"1.0","canonical_sha256":"476a3c706acd56a124967e4035063ece612eae3177ca08a9af475e721bba75b8","source":{"kind":"arxiv","id":"2606.24972","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.24972","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T11:45:11Z","cross_cats_sorted":[],"title_canon_sha256":"9742be0eebd79a0c9ac04541910f7147ce85d6bacc8dced5a36c2a46b04c9cbe","abstract_canon_sha256":"e811099156d582f988df4ea4763a6a035e9fa7886d794b99235bf0d5f7e29df8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-25T00:17:47.234975Z","signature_b64":"EQdc3XRxsVeM0VvAEkQzzzxbn8xRTsoKt7xMttCtXnFw7hd/srqp9DRKZ5f/R7627e0QMKAEAc0FgvlwMvwcAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"476a3c706acd56a124967e4035063ece612eae3177ca08a9af475e721bba75b8","last_reissued_at":"2026-06-25T00:17:47.234573Z","signature_status":"signed_v1","first_computed_at":"2026-06-25T00:17:47.234573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.24972","created_at":"2026-06-25T00:17:47.234637+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.24972v1","created_at":"2026-06-25T00:17:47.234637+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24972","created_at":"2026-06-25T00:17:47.234637+00:00"},{"alias_kind":"pith_short_12","alias_value":"I5VDY4DKZVLK","created_at":"2026-06-25T00:17:47.234637+00:00"},{"alias_kind":"pith_short_16","alias_value":"I5VDY4DKZVLKCJEW","created_at":"2026-06-25T00:17:47.234637+00:00"},{"alias_kind":"pith_short_8","alias_value":"I5VDY4DK","created_at":"2026-06-25T00:17:47.234637+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ","json":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ.json","graph_json":"https://pith.science/api/pith-number/I5VDY4DKZVLKCJEWPZADKBR6ZZ/graph.json","events_json":"https://pith.science/api/pith-number/I5VDY4DKZVLKCJEWPZADKBR6ZZ/events.json","paper":"https://pith.science/paper/I5VDY4DK"},"agent_actions":{"view_html":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ","download_json":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ.json","view_paper":"https://pith.science/paper/I5VDY4DK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.24972&json=true","fetch_graph":"https://pith.science/api/pith-number/I5VDY4DKZVLKCJEWPZADKBR6ZZ/graph.json","fetch_events":"https://pith.science/api/pith-number/I5VDY4DKZVLKCJEWPZADKBR6ZZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ/action/storage_attestation","attest_author":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ/action/author_attestation","sign_citation":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ/action/citation_signature","submit_replication":"https://pith.science/pith/I5VDY4DKZVLKCJEWPZADKBR6ZZ/action/replication_record"}},"created_at":"2026-06-25T00:17:47.234637+00:00","updated_at":"2026-06-25T00:17:47.234637+00:00"}