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In this paper we prove that $$\\frac{1}{2\\pi} \\int_0^{2\\pi}{\\left| (P_n - P_n^*)(e^{it}) \\right|^q \\, dt} \\sim \\frac{{2}^q \\Gamma \\left(\\frac{q+1}{2} \\right)}{\\Gamma \\left(\\frac q2 + 1 \\right) \\sqrt{\\pi}} \\,\\, n^{q/2}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \\in {\\mathcal K}_n$ and for ev"},"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":"1810.04287","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-09T22:40:23Z","cross_cats_sorted":[],"title_canon_sha256":"37f3284243fbc514d3060c623cda2121a942c7a41b0c3bd51e76829febdd24a2","abstract_canon_sha256":"ec9f60769c56aec3216fec353e95da84771a1fcd7f311e1dd2fdadc60e5084f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:15.977124Z","signature_b64":"VIW/WHHxyuaub+s7eKkKHMPxdzdU/QtPIeax8u3e3LjUhjkPMnS/21PB9xLvBddWuj3PMhVpst5Y+SI0AjU8Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86aac7b602517515fdaa2fe72469fdb5dda21b01f7ef6abb3384e60d8f220d48","last_reissued_at":"2026-05-17T23:54:15.976660Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:15.976660Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Tam\\'as Erd\\'elyi","submitted_at":"2018-10-09T22:40:23Z","abstract_excerpt":"Let $${\\mathcal K}_n := \\left\\{p_n: p_n(z) = \\sum_{k=0}^n{a_k z^k}, \\enspace a_k \\in {\\mathbb C}\\,,\\enspace |a_k| = 1 \\right\\}\\,.$$ A sequence $(P_n)$ of polynomials $P_n \\in {\\mathcal K}_n$ is called ultraflat if $(n + 1)^{-1/2}|P_n(e^{it})|$ converge to $1$ uniformly in $t \\in {\\mathbb R}$. In this paper we prove that $$\\frac{1}{2\\pi} \\int_0^{2\\pi}{\\left| (P_n - P_n^*)(e^{it}) \\right|^q \\, dt} \\sim \\frac{{2}^q \\Gamma \\left(\\frac{q+1}{2} \\right)}{\\Gamma \\left(\\frac q2 + 1 \\right) \\sqrt{\\pi}} \\,\\, n^{q/2}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \\in {\\mathcal K}_n$ and for ev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04287","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":""},"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":"1810.04287","created_at":"2026-05-17T23:54:15.976739+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.04287v2","created_at":"2026-05-17T23:54:15.976739+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.04287","created_at":"2026-05-17T23:54:15.976739+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q2VMPNQCKF2R","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q2VMPNQCKF2RL7NK","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q2VMPNQC","created_at":"2026-05-18T12:32:46.962924+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/Q2VMPNQCKF2RL7NKF7TSI2P5WX","json":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX.json","graph_json":"https://pith.science/api/pith-number/Q2VMPNQCKF2RL7NKF7TSI2P5WX/graph.json","events_json":"https://pith.science/api/pith-number/Q2VMPNQCKF2RL7NKF7TSI2P5WX/events.json","paper":"https://pith.science/paper/Q2VMPNQC"},"agent_actions":{"view_html":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX","download_json":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX.json","view_paper":"https://pith.science/paper/Q2VMPNQC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.04287&json=true","fetch_graph":"https://pith.science/api/pith-number/Q2VMPNQCKF2RL7NKF7TSI2P5WX/graph.json","fetch_events":"https://pith.science/api/pith-number/Q2VMPNQCKF2RL7NKF7TSI2P5WX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX/action/storage_attestation","attest_author":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX/action/author_attestation","sign_citation":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX/action/citation_signature","submit_replication":"https://pith.science/pith/Q2VMPNQCKF2RL7NKF7TSI2P5WX/action/replication_record"}},"created_at":"2026-05-17T23:54:15.976739+00:00","updated_at":"2026-05-17T23:54:15.976739+00:00"}