{"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"}