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Let $\\mathbf{P}_h$ be a uniform probability measure on the $L^2$ unit-sphere $S_h$ of this cluster of eigenfunctions and take $u \\in S_h$. Given a closed curve $\\gamma \\subset M$, there exists $C_{1}(\\gamma, M), C_{2}(\\gamma, M) > 0$ and $h_0>0$ such that for all $h \\in (0, h_0],$ \\begin{equation*}\n  C_1 h^{1/2} \\leq"},"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":"1401.1710","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-08T14:46:27Z","cross_cats_sorted":["math.PR","math.SP"],"title_canon_sha256":"43442b5ce399a85927ee6e33b42c70cdb5649e1a12ee1b67c679083ce6760aef","abstract_canon_sha256":"2d3089050f25fce271a5393f53313723a3f59a1775ff56a4322906ae632f7064"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:58.049623Z","signature_b64":"2B8sy4+ALUCH09CRiPJSUeBm6/EY2WZfQNsFNpTlHLHV2wyH9vQvROrJLjN4Py6zgFSlyONgEn78mGpBcw64AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5d4cb7b88d55c4cc0fa52ce51f0f2ae853b73f7003089a80eba01edc6ca90a1","last_reissued_at":"2026-05-18T03:02:58.048927Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:58.048927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Expected values of eigenfunction periods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","math.SP"],"primary_cat":"math.AP","authors_text":"Suresh Eswarathasan","submitted_at":"2014-01-08T14:46:27Z","abstract_excerpt":"Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2\\Delta_g \\phi_{h_j} = \\phi_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \\in (1, 1 + hD]$ for $D>0$ a large enough constant. Let $\\mathbf{P}_h$ be a uniform probability measure on the $L^2$ unit-sphere $S_h$ of this cluster of eigenfunctions and take $u \\in S_h$. Given a closed curve $\\gamma \\subset M$, there exists $C_{1}(\\gamma, M), C_{2}(\\gamma, M) > 0$ and $h_0>0$ such that for all $h \\in (0, h_0],$ \\begin{equation*}\n  C_1 h^{1/2} \\leq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1710","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":"1401.1710","created_at":"2026-05-18T03:02:58.049041+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.1710v1","created_at":"2026-05-18T03:02:58.049041+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.1710","created_at":"2026-05-18T03:02:58.049041+00:00"},{"alias_kind":"pith_short_12","alias_value":"4XKMW64I2VOE","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"4XKMW64I2VOEZQH2","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"4XKMW64I","created_at":"2026-05-18T12:28:14.216126+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/4XKMW64I2VOEZQH2KLHFD4HSV2","json":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2.json","graph_json":"https://pith.science/api/pith-number/4XKMW64I2VOEZQH2KLHFD4HSV2/graph.json","events_json":"https://pith.science/api/pith-number/4XKMW64I2VOEZQH2KLHFD4HSV2/events.json","paper":"https://pith.science/paper/4XKMW64I"},"agent_actions":{"view_html":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2","download_json":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2.json","view_paper":"https://pith.science/paper/4XKMW64I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.1710&json=true","fetch_graph":"https://pith.science/api/pith-number/4XKMW64I2VOEZQH2KLHFD4HSV2/graph.json","fetch_events":"https://pith.science/api/pith-number/4XKMW64I2VOEZQH2KLHFD4HSV2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2/action/storage_attestation","attest_author":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2/action/author_attestation","sign_citation":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2/action/citation_signature","submit_replication":"https://pith.science/pith/4XKMW64I2VOEZQH2KLHFD4HSV2/action/replication_record"}},"created_at":"2026-05-18T03:02:58.049041+00:00","updated_at":"2026-05-18T03:02:58.049041+00:00"}