On rate of convergence for universality limits

2023/06/23 by Roman Bessonov, Bessonov, Roman · 1 citation
Mathematics · #42C05 #46E22 #Analytic Number Theory Research #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2306.13722

openalex publication_date 2023/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a probability measure μ on the unit circle \mathbbT, consider the reproducing kernel k_μ,n(z₁, z₂) in the space of polynomials of degree at most n-1 with the L²(μ)-inner product. Let u, v ∈ ℂ. It is known that under mild assumptions on μ near ζ∈ \mathbbT, the ratio k_μ,n(ζe^u/n, ζe^v/n)/k_μ,n(ζ, ζ) converges to a universal limit S(u, v) as n → ∞. We give an estimate for the rate of this convergence for measures μ with finite logarithmic integral.

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