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We first prove that this IFS uniquely generates a Borel probability measure \\( \\mu \\). Furthermore, under the divisibility constraints \\[ p_2 \\mid b_2, \\quad 2 \\mid b_2, \\quad \\text{and} \\quad 2p_k \\mid b_k \\ \\"},"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":"2412.13427","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.FA","submitted_at":"2024-12-18T01:44:30Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"61a1d721c3b1f5e9e6c18a74597a4dbaf248fbb31b8174c836062c6a56e5a9f8","abstract_canon_sha256":"0d800b324497cb7a151d20724b348bdc3ab3e5f3cfc01d9acd276815088dd308"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:43:42.116052Z","signature_b64":"h9JiYs9ZIyVTPd5dPfGTik/0bor28Er8OPlVCioClynTKSad4VBA1KDABM0zMVzBQVyGXb9vfPo3BOcEPT1CBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78c95d2ad99fd9879e7c36c032f82d0ed1868c2997300769b5e8d1f0f3db2bc0","last_reissued_at":"2026-07-05T10:43:42.115547Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:43:42.115547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectrality of Moran-type measures with staggered contraction ratios","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Jing-Cheng Liu, Jun Jason Luo, Lin Mao","submitted_at":"2024-12-18T01:44:30Z","abstract_excerpt":"Consider a Moran-type iterated function system (IFS) \\( \\{\\phi_{k,d}\\}_{d\\in D_{2p_k}, k\\geq 1} \\), where each contraction map is defined as \\[ \\phi_{k,d}(x) = (-1)^d b_k^{-1}(x + d), \\] with integer sequences \\( \\{b_k\\}_{k=1}^\\infty \\) and \\( \\{p_k\\}_{k=1}^\\infty \\) satisfying \\( b_k \\geq 2p_k \\geq 2 \\), and digit sets \\( D_{2p_k} = \\{0, 1, \\ldots, 2p_k - 1\\} \\) for all \\( k \\geq 1 \\). We first prove that this IFS uniquely generates a Borel probability measure \\( \\mu \\). Furthermore, under the divisibility constraints \\[ p_2 \\mid b_2, \\quad 2 \\mid b_2, \\quad \\text{and} \\quad 2p_k \\mid b_k \\ \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.13427","kind":"arxiv","version":2},"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/2412.13427/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":"2412.13427","created_at":"2026-07-05T10:43:42.115609+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.13427v2","created_at":"2026-07-05T10:43:42.115609+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.13427","created_at":"2026-07-05T10:43:42.115609+00:00"},{"alias_kind":"pith_short_12","alias_value":"PDEV2KWZT7MY","created_at":"2026-07-05T10:43:42.115609+00:00"},{"alias_kind":"pith_short_16","alias_value":"PDEV2KWZT7MYPHT4","created_at":"2026-07-05T10:43:42.115609+00:00"},{"alias_kind":"pith_short_8","alias_value":"PDEV2KWZ","created_at":"2026-07-05T10:43:42.115609+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/PDEV2KWZT7MYPHT4G3ADF6BNB3","json":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3.json","graph_json":"https://pith.science/api/pith-number/PDEV2KWZT7MYPHT4G3ADF6BNB3/graph.json","events_json":"https://pith.science/api/pith-number/PDEV2KWZT7MYPHT4G3ADF6BNB3/events.json","paper":"https://pith.science/paper/PDEV2KWZ"},"agent_actions":{"view_html":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3","download_json":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3.json","view_paper":"https://pith.science/paper/PDEV2KWZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.13427&json=true","fetch_graph":"https://pith.science/api/pith-number/PDEV2KWZT7MYPHT4G3ADF6BNB3/graph.json","fetch_events":"https://pith.science/api/pith-number/PDEV2KWZT7MYPHT4G3ADF6BNB3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3/action/storage_attestation","attest_author":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3/action/author_attestation","sign_citation":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3/action/citation_signature","submit_replication":"https://pith.science/pith/PDEV2KWZT7MYPHT4G3ADF6BNB3/action/replication_record"}},"created_at":"2026-07-05T10:43:42.115609+00:00","updated_at":"2026-07-05T10:43:42.115609+00:00"}