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For $ 1 \\leq j \\leq d $, denote the $ j $-th the Lyapunov exponent by $ \\chi_{j} := \\sum_{i\\in\\Lambda} - p_{i} \\log | r_{i,j} |$, and define the IFS induced by $ \\Phi $ on the $j$-th coordinate as $ \\Phi_{j} := \\{ x \\mapsto r_{i,j}x + t_{i,j}\\}_{i\\in\\Lambda}$. We prove that if $ \\chi_{j_{1}} \\neq \\c"},"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":"2501.17378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-01-29T02:19:09Z","cross_cats_sorted":[],"title_canon_sha256":"dc240bbfc6a1bd0afc4833230954bfe7a1cfeae966f025ff5ce79e9a3ebedf4b","abstract_canon_sha256":"812bd40e05623ff3f6fa7003ea3b87f7771d32eddef2fd5192ef3dc4620db4cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:13:47.090299Z","signature_b64":"UFNW6DznKHqf76KNVeHoriTZgdBtG3HPIaxMQdBl2noT+YFpThY8JkY3n7oNbVf/oVBEVVO68jn0fsdytxsCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c7c3d6fc547023e07a3347392c3804194b9ca3335a9de52cc9f1ae4ae1651a4","last_reissued_at":"2026-07-05T10:13:47.089811Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:13:47.089811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dimension of diagonal self-affine measures with exponentially separated projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Zhou Feng","submitted_at":"2025-01-29T02:19:09Z","abstract_excerpt":"Let $ \\mu $ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $ \\Phi = \\{ (x_{1}, \\ldots, x_{d}) \\mapsto ( r_{i, 1}x_{1} + t_{i,1}, \\ldots, r_{i,d}x_{d} + t_{i,d}) \\}_{i\\in\\Lambda} $ on $ \\mathbb{R}^{d} $ and a probability vector $ p = (p_{i})_{i\\in\\Lambda}$. For $ 1 \\leq j \\leq d $, denote the $ j $-th the Lyapunov exponent by $ \\chi_{j} := \\sum_{i\\in\\Lambda} - p_{i} \\log | r_{i,j} |$, and define the IFS induced by $ \\Phi $ on the $j$-th coordinate as $ \\Phi_{j} := \\{ x \\mapsto r_{i,j}x + t_{i,j}\\}_{i\\in\\Lambda}$. We prove that if $ \\chi_{j_{1}} \\neq \\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.17378","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/2501.17378/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":"2501.17378","created_at":"2026-07-05T10:13:47.089864+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.17378v2","created_at":"2026-07-05T10:13:47.089864+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.17378","created_at":"2026-07-05T10:13:47.089864+00:00"},{"alias_kind":"pith_short_12","alias_value":"PR6D236FI4BD","created_at":"2026-07-05T10:13:47.089864+00:00"},{"alias_kind":"pith_short_16","alias_value":"PR6D236FI4BD4B5D","created_at":"2026-07-05T10:13:47.089864+00:00"},{"alias_kind":"pith_short_8","alias_value":"PR6D236F","created_at":"2026-07-05T10:13:47.089864+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2506.06851","citing_title":"Weakly separated self-affine carpets","ref_index":10,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG","json":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG.json","graph_json":"https://pith.science/api/pith-number/PR6D236FI4BD4B5DGRZZFQ4AIG/graph.json","events_json":"https://pith.science/api/pith-number/PR6D236FI4BD4B5DGRZZFQ4AIG/events.json","paper":"https://pith.science/paper/PR6D236F"},"agent_actions":{"view_html":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG","download_json":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG.json","view_paper":"https://pith.science/paper/PR6D236F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.17378&json=true","fetch_graph":"https://pith.science/api/pith-number/PR6D236FI4BD4B5DGRZZFQ4AIG/graph.json","fetch_events":"https://pith.science/api/pith-number/PR6D236FI4BD4B5DGRZZFQ4AIG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG/action/storage_attestation","attest_author":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG/action/author_attestation","sign_citation":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG/action/citation_signature","submit_replication":"https://pith.science/pith/PR6D236FI4BD4B5DGRZZFQ4AIG/action/replication_record"}},"created_at":"2026-07-05T10:13:47.089864+00:00","updated_at":"2026-07-05T10:13:47.089864+00:00"}