{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:55DXLYFZ6TLWIDGQ2HLSZFW3XU","short_pith_number":"pith:55DXLYFZ","schema_version":"1.0","canonical_sha256":"ef4775e0b9f4d7640cd0d1d72c96dbbd1ca9e806de4d9b0def18a02e60461075","source":{"kind":"arxiv","id":"1703.09553","version":1},"attestation_state":"computed","paper":{"title":"Patterns in Random Fractals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.PR","authors_text":"Pablo Shmerkin, Ville Suomala","submitted_at":"2017-03-28T12:56:14Z","abstract_excerpt":"We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer\\'{e}di theorems for random discrete sets, we also consider the corresponding problem for sets of positive $\\nu$-measure, where $\\nu$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investi"},"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":"1703.09553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-28T12:56:14Z","cross_cats_sorted":["math.CA","math.CO"],"title_canon_sha256":"cce4fc248543f998b2fbed520a1eab085ad65d373215df1e2e5bf5ebfe8caafb","abstract_canon_sha256":"f4e2353d1db1a44f361bcd87477b437b743de612428fc4b0051586d9ccdbe76d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:44.275652Z","signature_b64":"ob4kdVEWEp6OJcEG2HJFv955rwV6AR2WHMGdOvJcj0YzOeuoHBBlUoUWZ0NBhHKTNw8yV7J554dWAXeVZjKpDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef4775e0b9f4d7640cd0d1d72c96dbbd1ca9e806de4d9b0def18a02e60461075","last_reissued_at":"2026-05-18T00:47:44.274977Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:44.274977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Patterns in Random Fractals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.PR","authors_text":"Pablo Shmerkin, Ville Suomala","submitted_at":"2017-03-28T12:56:14Z","abstract_excerpt":"We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer\\'{e}di theorems for random discrete sets, we also consider the corresponding problem for sets of positive $\\nu$-measure, where $\\nu$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09553","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":"1703.09553","created_at":"2026-05-18T00:47:44.275087+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.09553v1","created_at":"2026-05-18T00:47:44.275087+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.09553","created_at":"2026-05-18T00:47:44.275087+00:00"},{"alias_kind":"pith_short_12","alias_value":"55DXLYFZ6TLW","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_16","alias_value":"55DXLYFZ6TLWIDGQ","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_8","alias_value":"55DXLYFZ","created_at":"2026-05-18T12:31:00.734936+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/55DXLYFZ6TLWIDGQ2HLSZFW3XU","json":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU.json","graph_json":"https://pith.science/api/pith-number/55DXLYFZ6TLWIDGQ2HLSZFW3XU/graph.json","events_json":"https://pith.science/api/pith-number/55DXLYFZ6TLWIDGQ2HLSZFW3XU/events.json","paper":"https://pith.science/paper/55DXLYFZ"},"agent_actions":{"view_html":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU","download_json":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU.json","view_paper":"https://pith.science/paper/55DXLYFZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.09553&json=true","fetch_graph":"https://pith.science/api/pith-number/55DXLYFZ6TLWIDGQ2HLSZFW3XU/graph.json","fetch_events":"https://pith.science/api/pith-number/55DXLYFZ6TLWIDGQ2HLSZFW3XU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU/action/storage_attestation","attest_author":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU/action/author_attestation","sign_citation":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU/action/citation_signature","submit_replication":"https://pith.science/pith/55DXLYFZ6TLWIDGQ2HLSZFW3XU/action/replication_record"}},"created_at":"2026-05-18T00:47:44.275087+00:00","updated_at":"2026-05-18T00:47:44.275087+00:00"}