{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:MW5RD63K3N4XZZDMX7KTOICC3B","short_pith_number":"pith:MW5RD63K","schema_version":"1.0","canonical_sha256":"65bb11fb6adb797ce46cbfd5372042d878ff22d3efd9a4172e8b16c02e405181","source":{"kind":"arxiv","id":"1307.6399","version":5},"attestation_state":"computed","paper":{"title":"Self similar sets, entropy and additive combinatorics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CA","authors_text":"Michael Hochman","submitted_at":"2013-07-24T12:18:54Z","abstract_excerpt":"This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in the original proof but also falls short of a complete proof. In the last section we discuss how the heuristic argument "},"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":"1307.6399","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-24T12:18:54Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"401042864fbcb8895636c8c54ba0efd267f56c51af6503f7e88062bf768b8219","abstract_canon_sha256":"5f567a4aba6f3b8bbef6fe8c2c7127b8b45f39c652752141609e071c24f8ef69"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:42.272900Z","signature_b64":"zg2MmgPc9yg/BlhPP473a3xqu7r/RTrD7JR60oMq2IfjfxO/I8G0incwZZ2yCbJSlEAzEH5LfJ1q65aF4E5eDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65bb11fb6adb797ce46cbfd5372042d878ff22d3efd9a4172e8b16c02e405181","last_reissued_at":"2026-05-18T02:41:42.272242Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:42.272242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self similar sets, entropy and additive combinatorics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CA","authors_text":"Michael Hochman","submitted_at":"2013-07-24T12:18:54Z","abstract_excerpt":"This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in the original proof but also falls short of a complete proof. In the last section we discuss how the heuristic argument "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6399","kind":"arxiv","version":5},"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":"1307.6399","created_at":"2026-05-18T02:41:42.272334+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.6399v5","created_at":"2026-05-18T02:41:42.272334+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.6399","created_at":"2026-05-18T02:41:42.272334+00:00"},{"alias_kind":"pith_short_12","alias_value":"MW5RD63K3N4X","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"MW5RD63K3N4XZZDM","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"MW5RD63K","created_at":"2026-05-18T12:27:52.871228+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/MW5RD63K3N4XZZDMX7KTOICC3B","json":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B.json","graph_json":"https://pith.science/api/pith-number/MW5RD63K3N4XZZDMX7KTOICC3B/graph.json","events_json":"https://pith.science/api/pith-number/MW5RD63K3N4XZZDMX7KTOICC3B/events.json","paper":"https://pith.science/paper/MW5RD63K"},"agent_actions":{"view_html":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B","download_json":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B.json","view_paper":"https://pith.science/paper/MW5RD63K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.6399&json=true","fetch_graph":"https://pith.science/api/pith-number/MW5RD63K3N4XZZDMX7KTOICC3B/graph.json","fetch_events":"https://pith.science/api/pith-number/MW5RD63K3N4XZZDMX7KTOICC3B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B/action/storage_attestation","attest_author":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B/action/author_attestation","sign_citation":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B/action/citation_signature","submit_replication":"https://pith.science/pith/MW5RD63K3N4XZZDMX7KTOICC3B/action/replication_record"}},"created_at":"2026-05-18T02:41:42.272334+00:00","updated_at":"2026-05-18T02:41:42.272334+00:00"}