{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:XZXVQ2XDPEBDMME6XLA5V6FUEZ","short_pith_number":"pith:XZXVQ2XD","schema_version":"1.0","canonical_sha256":"be6f586ae3790236309ebac1daf8b4266c0eb5181372164cc86c5c89c8361a73","source":{"kind":"arxiv","id":"1103.3088","version":1},"attestation_state":"computed","paper":{"title":"Optimal Discrete Riesz Energy and Discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"J. S. Brauchart","submitted_at":"2011-03-16T03:52:35Z","abstract_excerpt":"The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the Euclidean distance) governing the point interaction is replaced with the logarithmic potential $\\log(1/r)$. In particular, we present a conjecture for the leading term of the asymptotic expansion of the optimal $\\IL_2$-discrepancy with respect to spherical caps on the unit sphere in $\\mathbb{R}^{d+1}$ which follows from Stolarsky's invariance principle [Proc"},"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":"1103.3088","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-03-16T03:52:35Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"63ae735ca20406a129d6e0534a854d5ef2720055556e0459440e0dabc5a3e82a","abstract_canon_sha256":"7798a2ce448be7afe3bbb545e1ab14f0944f0513d7ef75080c357740c148ec64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:06.742875Z","signature_b64":"ECeNZ/oN1CoVew1J9bf+78kq0VDtTWzBdlWmgtKdeQL3DT8nsRxUE6cuxeEKZr0QZcMrixfoFwuz4UmFbRMXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be6f586ae3790236309ebac1daf8b4266c0eb5181372164cc86c5c89c8361a73","last_reissued_at":"2026-05-18T02:59:06.742089Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:06.742089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Discrete Riesz Energy and Discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"J. S. Brauchart","submitted_at":"2011-03-16T03:52:35Z","abstract_excerpt":"The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the Euclidean distance) governing the point interaction is replaced with the logarithmic potential $\\log(1/r)$. In particular, we present a conjecture for the leading term of the asymptotic expansion of the optimal $\\IL_2$-discrepancy with respect to spherical caps on the unit sphere in $\\mathbb{R}^{d+1}$ which follows from Stolarsky's invariance principle [Proc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3088","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":"1103.3088","created_at":"2026-05-18T02:59:06.742225+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.3088v1","created_at":"2026-05-18T02:59:06.742225+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3088","created_at":"2026-05-18T02:59:06.742225+00:00"},{"alias_kind":"pith_short_12","alias_value":"XZXVQ2XDPEBD","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"XZXVQ2XDPEBDMME6","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"XZXVQ2XD","created_at":"2026-05-18T12:26:47.523578+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/XZXVQ2XDPEBDMME6XLA5V6FUEZ","json":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ.json","graph_json":"https://pith.science/api/pith-number/XZXVQ2XDPEBDMME6XLA5V6FUEZ/graph.json","events_json":"https://pith.science/api/pith-number/XZXVQ2XDPEBDMME6XLA5V6FUEZ/events.json","paper":"https://pith.science/paper/XZXVQ2XD"},"agent_actions":{"view_html":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ","download_json":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ.json","view_paper":"https://pith.science/paper/XZXVQ2XD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.3088&json=true","fetch_graph":"https://pith.science/api/pith-number/XZXVQ2XDPEBDMME6XLA5V6FUEZ/graph.json","fetch_events":"https://pith.science/api/pith-number/XZXVQ2XDPEBDMME6XLA5V6FUEZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ/action/storage_attestation","attest_author":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ/action/author_attestation","sign_citation":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ/action/citation_signature","submit_replication":"https://pith.science/pith/XZXVQ2XDPEBDMME6XLA5V6FUEZ/action/replication_record"}},"created_at":"2026-05-18T02:59:06.742225+00:00","updated_at":"2026-05-18T02:59:06.742225+00:00"}