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When $m$ is odd, we show that the variation $|\\nu_\\beta^{(n)}|$ coincides the unsigned Bernoulli convolution $$\\mu_\\beta^{(n)}=*_{j=1}^n \\left (\\frac12\\delta_{\\beta^{-j}}+\\frac12\\delta_{-\\beta^{-j}}\\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\\|\\nu_\\beta^{(n)}\\|$ as $n\\rightarrow\\infty$."},"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":"1710.01780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-04T19:44:03Z","cross_cats_sorted":["math.CO","math.DS"],"title_canon_sha256":"cb0ce5dcd839d8277d6d863b266d01fa1f554f8f60b1b7cbd1b9869e9d8ed757","abstract_canon_sha256":"13cb29ae7c80ba752e82fb43fddcffd65c653da1df34d37ec97b45f6066a8127"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:38.524250Z","signature_b64":"ogXlpHwDGircFkXVmozxryIsMcBMDSNtUevaJy2rnnCcssw1sDjMIb825X2aU7DXU9ZHLPBPBq/MPmgAZgd9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb49fe827001d93afb21c6fe069b91b16404d39a800f63978636ce4c3c6855b3","last_reissued_at":"2026-05-18T00:33:38.523746Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:38.523746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of signed Bernoulli convolutions scaled by multinacci numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.DS"],"primary_cat":"math.CA","authors_text":"Tian-You Hu, Xianghong Chen","submitted_at":"2017-10-04T19:44:03Z","abstract_excerpt":"We study the signed Bernoulli convolution $$\\nu_\\beta^{(n)}=*_{j=1}^n \\left (\\frac12\\delta_{\\beta^{-j}}-\\frac12\\delta_{-\\beta^{-j}}\\right ),\\ n\\ge 1$$ where $\\beta>1$ satisfies $$\\beta^m=\\beta^{m-1}+\\cdots+\\beta+1$$ for some integer $m\\ge 2$. When $m$ is odd, we show that the variation $|\\nu_\\beta^{(n)}|$ coincides the unsigned Bernoulli convolution $$\\mu_\\beta^{(n)}=*_{j=1}^n \\left (\\frac12\\delta_{\\beta^{-j}}+\\frac12\\delta_{-\\beta^{-j}}\\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\\|\\nu_\\beta^{(n)}\\|$ as $n\\rightarrow\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01780","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":"1710.01780","created_at":"2026-05-18T00:33:38.523827+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.01780v1","created_at":"2026-05-18T00:33:38.523827+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.01780","created_at":"2026-05-18T00:33:38.523827+00:00"},{"alias_kind":"pith_short_12","alias_value":"5NE75ATQAHMT","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_16","alias_value":"5NE75ATQAHMTV6ZB","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_8","alias_value":"5NE75ATQ","created_at":"2026-05-18T12:31:00.734936+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/5NE75ATQAHMTV6ZBY37ANG4RWF","json":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF.json","graph_json":"https://pith.science/api/pith-number/5NE75ATQAHMTV6ZBY37ANG4RWF/graph.json","events_json":"https://pith.science/api/pith-number/5NE75ATQAHMTV6ZBY37ANG4RWF/events.json","paper":"https://pith.science/paper/5NE75ATQ"},"agent_actions":{"view_html":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF","download_json":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF.json","view_paper":"https://pith.science/paper/5NE75ATQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.01780&json=true","fetch_graph":"https://pith.science/api/pith-number/5NE75ATQAHMTV6ZBY37ANG4RWF/graph.json","fetch_events":"https://pith.science/api/pith-number/5NE75ATQAHMTV6ZBY37ANG4RWF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF/action/storage_attestation","attest_author":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF/action/author_attestation","sign_citation":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF/action/citation_signature","submit_replication":"https://pith.science/pith/5NE75ATQAHMTV6ZBY37ANG4RWF/action/replication_record"}},"created_at":"2026-05-18T00:33:38.523827+00:00","updated_at":"2026-05-18T00:33:38.523827+00:00"}