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More generally, given two self-similar sets $K,K'$ in $\\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\\dim(K)+\\dim(K') \\le 1$ (``geometric resonance''), then there exists $r<1$ such that all contrac"},"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":"0705.2628","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2007-05-18T04:08:27Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f10b3d18b3ee1c72e88ee1ecbd79baccb9cd915533cbd01aee542bb52d0be259","abstract_canon_sha256":"9fe7351fa018f42e9df060a05365efd008f05a61899e9b4c862753e2d716990f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:26.389859Z","signature_b64":"9BYrJgWHnS7kFGlHabGPAZQZ6g68WMdGwrt/eopc6p08lRlo9KQtVVNal0JAZ/0bRuihbxihAYgGhVXZBGeuBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95400b6636c3cc4a379703d2da7b0cf160e3d5d9f9ffa22137cd0b55681dffa3","last_reissued_at":"2026-05-18T03:30:26.389111Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:26.389111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Resonance between Cantor sets","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CA","authors_text":"Pablo Shmerkin, Yuval Peres","submitted_at":"2007-05-18T04:08:27Z","abstract_excerpt":"Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\\log b/\\log a$ is irrational, then \\[ \\dim(C_a+C_b) = \\min(\\dim(C_a) + \\dim(C_b),1), \\] where $\\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\\dim(K)+\\dim(K') \\le 1$ (``geometric resonance''), then there exists $r<1$ such that all contrac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.2628","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":""},"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":"0705.2628","created_at":"2026-05-18T03:30:26.389250+00:00"},{"alias_kind":"arxiv_version","alias_value":"0705.2628v2","created_at":"2026-05-18T03:30:26.389250+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0705.2628","created_at":"2026-05-18T03:30:26.389250+00:00"},{"alias_kind":"pith_short_12","alias_value":"SVAAWZRWYPGE","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"SVAAWZRWYPGEUN4X","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"SVAAWZRW","created_at":"2026-05-18T12:25:56.245647+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/SVAAWZRWYPGEUN4XAPJNU6YM6F","json":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F.json","graph_json":"https://pith.science/api/pith-number/SVAAWZRWYPGEUN4XAPJNU6YM6F/graph.json","events_json":"https://pith.science/api/pith-number/SVAAWZRWYPGEUN4XAPJNU6YM6F/events.json","paper":"https://pith.science/paper/SVAAWZRW"},"agent_actions":{"view_html":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F","download_json":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F.json","view_paper":"https://pith.science/paper/SVAAWZRW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0705.2628&json=true","fetch_graph":"https://pith.science/api/pith-number/SVAAWZRWYPGEUN4XAPJNU6YM6F/graph.json","fetch_events":"https://pith.science/api/pith-number/SVAAWZRWYPGEUN4XAPJNU6YM6F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F/action/storage_attestation","attest_author":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F/action/author_attestation","sign_citation":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F/action/citation_signature","submit_replication":"https://pith.science/pith/SVAAWZRWYPGEUN4XAPJNU6YM6F/action/replication_record"}},"created_at":"2026-05-18T03:30:26.389250+00:00","updated_at":"2026-05-18T03:30:26.389250+00:00"}