{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:UW5AE6HOI33YCUIPFNE6YIQLXZ","short_pith_number":"pith:UW5AE6HO","schema_version":"1.0","canonical_sha256":"a5ba0278ee46f781510f2b49ec220bbe5091218d4265fa364121d0f9b2b768ec","source":{"kind":"arxiv","id":"2201.02059","version":1},"attestation_state":"computed","paper":{"title":"On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.DS","authors_text":"Yiftach Dayan","submitted_at":"2022-01-06T13:50:40Z","abstract_excerpt":"We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring distribution is not necessarily binomial) are projected to $\\mathbb{R}^d$ by a coding map which arises from an iterated function system (IFS) of similarity maps. We show that for such a random fractal $E$, whenever the underlying IFS satisfies the open set condition, almost surely the Assouad dimension of $E$ is the maximal Hausdorff dimension of a set in $\\text{"},"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":"2201.02059","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2022-01-06T13:50:40Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"d711086f9d854eb721f2eec257b369ad9b005a4bfba0f9d566ccf2a85d084679","abstract_canon_sha256":"a8b66784c8e7c71e1cb80caf8efa041a1726f73be79197a20ef585921d8a06b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:46:29.476404Z","signature_b64":"tAPe4kT/gLa0+6i/RkAaQ7uHKSGb9sozutaar2kxuIygecCv6ktEia/lVFgzOfOHzNFQh4bzMxmJ49o1WOAvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5ba0278ee46f781510f2b49ec220bbe5091218d4265fa364121d0f9b2b768ec","last_reissued_at":"2026-07-05T03:46:29.475979Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:46:29.475979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.DS","authors_text":"Yiftach Dayan","submitted_at":"2022-01-06T13:50:40Z","abstract_excerpt":"We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring distribution is not necessarily binomial) are projected to $\\mathbb{R}^d$ by a coding map which arises from an iterated function system (IFS) of similarity maps. We show that for such a random fractal $E$, whenever the underlying IFS satisfies the open set condition, almost surely the Assouad dimension of $E$ is the maximal Hausdorff dimension of a set in $\\text{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.02059","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.02059/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":"2201.02059","created_at":"2026-07-05T03:46:29.476046+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.02059v1","created_at":"2026-07-05T03:46:29.476046+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.02059","created_at":"2026-07-05T03:46:29.476046+00:00"},{"alias_kind":"pith_short_12","alias_value":"UW5AE6HOI33Y","created_at":"2026-07-05T03:46:29.476046+00:00"},{"alias_kind":"pith_short_16","alias_value":"UW5AE6HOI33YCUIP","created_at":"2026-07-05T03:46:29.476046+00:00"},{"alias_kind":"pith_short_8","alias_value":"UW5AE6HO","created_at":"2026-07-05T03:46:29.476046+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.27869","citing_title":"Walk dimension and vanishing curve modulus in metric measure spaces","ref_index":7,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ","json":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ.json","graph_json":"https://pith.science/api/pith-number/UW5AE6HOI33YCUIPFNE6YIQLXZ/graph.json","events_json":"https://pith.science/api/pith-number/UW5AE6HOI33YCUIPFNE6YIQLXZ/events.json","paper":"https://pith.science/paper/UW5AE6HO"},"agent_actions":{"view_html":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ","download_json":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ.json","view_paper":"https://pith.science/paper/UW5AE6HO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.02059&json=true","fetch_graph":"https://pith.science/api/pith-number/UW5AE6HOI33YCUIPFNE6YIQLXZ/graph.json","fetch_events":"https://pith.science/api/pith-number/UW5AE6HOI33YCUIPFNE6YIQLXZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ/action/storage_attestation","attest_author":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ/action/author_attestation","sign_citation":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ/action/citation_signature","submit_replication":"https://pith.science/pith/UW5AE6HOI33YCUIPFNE6YIQLXZ/action/replication_record"}},"created_at":"2026-07-05T03:46:29.476046+00:00","updated_at":"2026-07-05T03:46:29.476046+00:00"}