{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:XERNMN6ZTN46CJGOCBHJA33N4B","short_pith_number":"pith:XERNMN6Z","schema_version":"1.0","canonical_sha256":"b922d637d99b79e124ce104e906f6de04d364c81e08a120ded2ef0c306dd0d66","source":{"kind":"arxiv","id":"2507.21020","version":2},"attestation_state":"computed","paper":{"title":"Medians, Oscillations, and Distance Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ignacio Uriarte-Tuero, Marcus Pasquariello","submitted_at":"2025-07-28T17:42:19Z","abstract_excerpt":"Vasin (for $n=1$) and Anderson, Lehrb\\\"ack, Mudarra, and V\\\"ah\\\"akangas (arXiv:2209.06284) (for $n>1$) provided a geometric characterization of the sets $E \\subset \\mathbb{R}^n$ so that $w = \\text{dist}(\\cdot, E)^{-\\alpha}$ is a Muckenhoupt $A_1$ weight for some $\\alpha > 0$. In this paper, we provide a geometric characterization of the sets $E \\subset \\mathbb{R}^n$ (which we call median porous sets) so that $w = \\text{dist}(\\cdot, E)^{-\\alpha}$ is a Muckenhoupt $A_p$ weight for some $\\alpha > 0$ (given any $1 < p \\leq \\infty$).\n  Given $1 < p \\leq \\infty$, we also find the precise range of ex"},"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":"2507.21020","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2025-07-28T17:42:19Z","cross_cats_sorted":[],"title_canon_sha256":"9bd39312361cac8de8f6d3d1dd2635d7cc1be14848d62998b68d4d22b37de263","abstract_canon_sha256":"13f5681e2cb371e831b140adfedcb12aa5d82afc73ceb9c75a6ec8a17e632809"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:05:19.153199Z","signature_b64":"cl7bek3E68R2lFs+Js1as6G2IMdswdJ+D2O/r1932AMiLwL9EP490YEVG6dOb4kubaODt0BUEcexfy2T+D3FBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b922d637d99b79e124ce104e906f6de04d364c81e08a120ded2ef0c306dd0d66","last_reissued_at":"2026-07-05T12:05:19.152656Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:05:19.152656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Medians, Oscillations, and Distance Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ignacio Uriarte-Tuero, Marcus Pasquariello","submitted_at":"2025-07-28T17:42:19Z","abstract_excerpt":"Vasin (for $n=1$) and Anderson, Lehrb\\\"ack, Mudarra, and V\\\"ah\\\"akangas (arXiv:2209.06284) (for $n>1$) provided a geometric characterization of the sets $E \\subset \\mathbb{R}^n$ so that $w = \\text{dist}(\\cdot, E)^{-\\alpha}$ is a Muckenhoupt $A_1$ weight for some $\\alpha > 0$. In this paper, we provide a geometric characterization of the sets $E \\subset \\mathbb{R}^n$ (which we call median porous sets) so that $w = \\text{dist}(\\cdot, E)^{-\\alpha}$ is a Muckenhoupt $A_p$ weight for some $\\alpha > 0$ (given any $1 < p \\leq \\infty$).\n  Given $1 < p \\leq \\infty$, we also find the precise range of ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.21020","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/2507.21020/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":"2507.21020","created_at":"2026-07-05T12:05:19.152715+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.21020v2","created_at":"2026-07-05T12:05:19.152715+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.21020","created_at":"2026-07-05T12:05:19.152715+00:00"},{"alias_kind":"pith_short_12","alias_value":"XERNMN6ZTN46","created_at":"2026-07-05T12:05:19.152715+00:00"},{"alias_kind":"pith_short_16","alias_value":"XERNMN6ZTN46CJGO","created_at":"2026-07-05T12:05:19.152715+00:00"},{"alias_kind":"pith_short_8","alias_value":"XERNMN6Z","created_at":"2026-07-05T12:05:19.152715+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.05034","citing_title":"Median porosity is quasiconformally invariant","ref_index":11,"is_internal_anchor":false},{"citing_arxiv_id":"2607.01167","citing_title":"One-sided median porous sets and one-sided Muckenhoupt distance functions","ref_index":21,"is_internal_anchor":false},{"citing_arxiv_id":"2604.12561","citing_title":"Parabolic weak porosity and parabolic Muckenhoupt distance functions","ref_index":23,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B","json":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B.json","graph_json":"https://pith.science/api/pith-number/XERNMN6ZTN46CJGOCBHJA33N4B/graph.json","events_json":"https://pith.science/api/pith-number/XERNMN6ZTN46CJGOCBHJA33N4B/events.json","paper":"https://pith.science/paper/XERNMN6Z"},"agent_actions":{"view_html":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B","download_json":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B.json","view_paper":"https://pith.science/paper/XERNMN6Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.21020&json=true","fetch_graph":"https://pith.science/api/pith-number/XERNMN6ZTN46CJGOCBHJA33N4B/graph.json","fetch_events":"https://pith.science/api/pith-number/XERNMN6ZTN46CJGOCBHJA33N4B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B/action/storage_attestation","attest_author":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B/action/author_attestation","sign_citation":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B/action/citation_signature","submit_replication":"https://pith.science/pith/XERNMN6ZTN46CJGOCBHJA33N4B/action/replication_record"}},"created_at":"2026-07-05T12:05:19.152715+00:00","updated_at":"2026-07-05T12:05:19.152715+00:00"}