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There are two natural densities associated to this problem: the discrepancy density $\\rho_{\\mathbb{H}}$, given by $$ \\rho_{\\mathbb{H}} = \\liminf_{r\\to 1^-} \\inf_{f} \\frac{\\int_{\\mathbb{D}(0,r)} \\left((1-\\lvert z\\rvert^2) \\lvert f(z)\\rvert-1\\right)^2 \\frac{dA(z)}{1-\\lvert z\\rvert^2}} {\\int_{\\mathbb{D}(0,r)} \\frac{dA(z)}{1-\\lvert z\\rvert^2}} $$ which measures the discrepancy in optimal approximation of $(1-\\lvert z\\rvert^2)^{-1}$ with the modulus of polynomials $f$, and it's relative, the tight discrepancy density "},"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":"1605.08674","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-05-27T14:49:34Z","cross_cats_sorted":[],"title_canon_sha256":"dfecaeac9d9d1d6a5557b0cbf0ca8ef19b204cac4e88f984c4560ef40402a848","abstract_canon_sha256":"0c974f39f4aeab473550b4f0dbe4c04308862301184ff2f65bab2a5222e91d55"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:29.351333Z","signature_b64":"6Q4gBm3FiPoMLPvzk4uPY1awOshI326Jn1ZYDjLamJzyyYNLbKTwRefME5R3I5IdV00UuMK+leP/UKFjHS+4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"595a597401849803615dd27ad19b0a7334b2237854da2d682fa303943bbdcd69","last_reissued_at":"2026-05-18T01:13:29.350711Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:29.350711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrepancy densities for planar and hyperbolic Zero Packing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Aron Wennman","submitted_at":"2016-05-27T14:49:34Z","abstract_excerpt":"We study the problem of geometric zero packing, recently introduced by Hedenmalm. There are two natural densities associated to this problem: the discrepancy density $\\rho_{\\mathbb{H}}$, given by $$ \\rho_{\\mathbb{H}} = \\liminf_{r\\to 1^-} \\inf_{f} \\frac{\\int_{\\mathbb{D}(0,r)} \\left((1-\\lvert z\\rvert^2) \\lvert f(z)\\rvert-1\\right)^2 \\frac{dA(z)}{1-\\lvert z\\rvert^2}} {\\int_{\\mathbb{D}(0,r)} \\frac{dA(z)}{1-\\lvert z\\rvert^2}} $$ which measures the discrepancy in optimal approximation of $(1-\\lvert z\\rvert^2)^{-1}$ with the modulus of polynomials $f$, and it's relative, the tight discrepancy density "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.08674","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":"1605.08674","created_at":"2026-05-18T01:13:29.350791+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.08674v1","created_at":"2026-05-18T01:13:29.350791+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.08674","created_at":"2026-05-18T01:13:29.350791+00:00"},{"alias_kind":"pith_short_12","alias_value":"LFNFS5ABQSMA","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LFNFS5ABQSMAGYK5","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LFNFS5AB","created_at":"2026-05-18T12:30:29.479603+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/LFNFS5ABQSMAGYK52J5NDGYKOM","json":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM.json","graph_json":"https://pith.science/api/pith-number/LFNFS5ABQSMAGYK52J5NDGYKOM/graph.json","events_json":"https://pith.science/api/pith-number/LFNFS5ABQSMAGYK52J5NDGYKOM/events.json","paper":"https://pith.science/paper/LFNFS5AB"},"agent_actions":{"view_html":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM","download_json":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM.json","view_paper":"https://pith.science/paper/LFNFS5AB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.08674&json=true","fetch_graph":"https://pith.science/api/pith-number/LFNFS5ABQSMAGYK52J5NDGYKOM/graph.json","fetch_events":"https://pith.science/api/pith-number/LFNFS5ABQSMAGYK52J5NDGYKOM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM/action/storage_attestation","attest_author":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM/action/author_attestation","sign_citation":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM/action/citation_signature","submit_replication":"https://pith.science/pith/LFNFS5ABQSMAGYK52J5NDGYKOM/action/replication_record"}},"created_at":"2026-05-18T01:13:29.350791+00:00","updated_at":"2026-05-18T01:13:29.350791+00:00"}