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We show that if $\\det(M)\\notin 3\\mathbb{Z}$, then the mutually orthogonal exponential functions in $L^2(\\mu_{M,D})$ is finite, and the exact maximal cardinality is given."},"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":"1805.09941","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-25T00:56:31Z","cross_cats_sorted":[],"title_canon_sha256":"e68bafd5b9034d298221dab2c36ccc309cf1892eb2b191c5aa3945867afb3591","abstract_canon_sha256":"fb0506f503bdca5ad312fcd33217745576cef13fdac3919e5592168a22a8fafb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:58.737708Z","signature_b64":"YCh89RbN4QXRPMPsaPCHzKoUgWBRm/5OQNITqyFUgT4rrU3K8dOu/UE945dvXioBagU+GBqsIJ0Yzf5L2JjtDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a485d6873089c0f87e8bd6711b83a9110c2a63c61549ecfda37d63d646d6e197","last_reissued_at":"2026-05-18T00:14:58.737014Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:58.737014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jing-Cheng Liu, Ming-Liang Chen","submitted_at":"2018-05-25T00:56:31Z","abstract_excerpt":"In this paper, we consider the planar self-affine measures $\\mu_{M,D}$ generated by an expanding matrix $M\\in M_2(\\mathbb{Z})$ and an integer digit set $\nD=\\left\\{ {\\left( {\\begin{array}{*{20}{c}} 0\\\\ 0\n\\end{array}} \\right),\\left( {\\begin{array}{*{20}{c}} \\alpha_1\\\\ \\alpha_2 \\end{array}} \\right),\\left( {\\begin{array}{*{20}{c}} \\beta_1\\\\ \\beta_2 \\end{array}} \\right)} \\right\\} $\nwith $\\alpha_1\\beta_2-\\alpha_2\\beta_1\\neq0$. We show that if $\\det(M)\\notin 3\\mathbb{Z}$, then the mutually orthogonal exponential functions in $L^2(\\mu_{M,D})$ is finite, and the exact maximal cardinality is given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09941","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":"1805.09941","created_at":"2026-05-18T00:14:58.737129+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.09941v1","created_at":"2026-05-18T00:14:58.737129+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09941","created_at":"2026-05-18T00:14:58.737129+00:00"},{"alias_kind":"pith_short_12","alias_value":"USC5NBZQRHAP","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"USC5NBZQRHAPQ7UL","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"USC5NBZQ","created_at":"2026-05-18T12:32:56.356000+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/USC5NBZQRHAPQ7UL2ZYRXA5JCE","json":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE.json","graph_json":"https://pith.science/api/pith-number/USC5NBZQRHAPQ7UL2ZYRXA5JCE/graph.json","events_json":"https://pith.science/api/pith-number/USC5NBZQRHAPQ7UL2ZYRXA5JCE/events.json","paper":"https://pith.science/paper/USC5NBZQ"},"agent_actions":{"view_html":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE","download_json":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE.json","view_paper":"https://pith.science/paper/USC5NBZQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.09941&json=true","fetch_graph":"https://pith.science/api/pith-number/USC5NBZQRHAPQ7UL2ZYRXA5JCE/graph.json","fetch_events":"https://pith.science/api/pith-number/USC5NBZQRHAPQ7UL2ZYRXA5JCE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE/action/storage_attestation","attest_author":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE/action/author_attestation","sign_citation":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE/action/citation_signature","submit_replication":"https://pith.science/pith/USC5NBZQRHAPQ7UL2ZYRXA5JCE/action/replication_record"}},"created_at":"2026-05-18T00:14:58.737129+00:00","updated_at":"2026-05-18T00:14:58.737129+00:00"}