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The special case is the Cantor measure with $\\rho =\\frac 1{2k}$ and $N=2$ \\cite {JP}, which was proved recently to be the only spectral measure among the Bernoulli convolutions with $0<\\rho<1$ \\cite {D}."},"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":"1403.0669","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-03-04T02:54:57Z","cross_cats_sorted":[],"title_canon_sha256":"636a7494058082c0acfdc01cd6b4ae28e1a15c5b1ff4b876f7d28f91bf833cd7","abstract_canon_sha256":"9648245d51b81c1824ba05833dd2e7decba5711d565e89f043e09c2f3d809373"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:15.999406Z","signature_b64":"YEhTMsTd1oQ9OQiTlEHNFy2TUea1bKjlNKCmlqQse14baonUJhInAH2/n2nsCqi2INwoMQLja1g1KTSxyPbmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f9604d03275d86fe9f11335946cee531a0114a47b50507a7bc673eaabcdc028","last_reissued_at":"2026-05-18T02:57:15.998773Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:15.998773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Spectral N-Bernoulli Mmeasure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ka-Sing Lau, Xing-Gang He, Xin-Rong Dai","submitted_at":"2014-03-04T02:54:57Z","abstract_excerpt":"For $0<\\rho<1$ and $N>1$ an integer, let $\\mu$ be the self-similar measure defined by $\\mu(\\cdot)=\\sum_{i=0}^{N-1}\\frac 1N\\mu(\\rho^{-1}(\\cdot)-i)$. We prove that $L^2(\\mu)$ has an exponential orthonormal basis if and only if $\\rho=\\frac 1q$ for some $q>0$ and $N$ divides $q$. The special case is the Cantor measure with $\\rho =\\frac 1{2k}$ and $N=2$ \\cite {JP}, which was proved recently to be the only spectral measure among the Bernoulli convolutions with $0<\\rho<1$ \\cite {D}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0669","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":""},"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":"1403.0669","created_at":"2026-05-18T02:57:15.998858+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.0669v1","created_at":"2026-05-18T02:57:15.998858+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.0669","created_at":"2026-05-18T02:57:15.998858+00:00"},{"alias_kind":"pith_short_12","alias_value":"F6LAJUBSOXMG","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"F6LAJUBSOXMG72PR","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"F6LAJUBS","created_at":"2026-05-18T12:28:28.263976+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/F6LAJUBSOXMG72PRCM2ZI3HOKM","json":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM.json","graph_json":"https://pith.science/api/pith-number/F6LAJUBSOXMG72PRCM2ZI3HOKM/graph.json","events_json":"https://pith.science/api/pith-number/F6LAJUBSOXMG72PRCM2ZI3HOKM/events.json","paper":"https://pith.science/paper/F6LAJUBS"},"agent_actions":{"view_html":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM","download_json":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM.json","view_paper":"https://pith.science/paper/F6LAJUBS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.0669&json=true","fetch_graph":"https://pith.science/api/pith-number/F6LAJUBSOXMG72PRCM2ZI3HOKM/graph.json","fetch_events":"https://pith.science/api/pith-number/F6LAJUBSOXMG72PRCM2ZI3HOKM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM/action/storage_attestation","attest_author":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM/action/author_attestation","sign_citation":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM/action/citation_signature","submit_replication":"https://pith.science/pith/F6LAJUBSOXMG72PRCM2ZI3HOKM/action/replication_record"}},"created_at":"2026-05-18T02:57:15.998858+00:00","updated_at":"2026-05-18T02:57:15.998858+00:00"}