---
abstract: 'We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod $p$ and only finitely many of them.'
author:
- 'Barry Simon\*'
date: 'December 16, 2004'
title: |
    Fine Structure of the Zeros of Orthogonal Polynomials,\
    III. Periodic Recursion Coefficients
---

[^1] [^2]

Introduction
============

[s1]{} This paper is the third in a series [@SaffProc; @Saff2] that discusses detailed asymptotics of the zeros of orthogonal polynomials with special emphasis on distances between nearby zeros. We discuss both orthogonal polynomials on the real line (OPRL) where the basic recursion for the orthonormal polynomials, $p_n(x)$, is $${\label}{1.1} 
xp_n(x) = a_{n+1} p_{n+1}(x) + b_{n+1} p_n (x) + a_n p_{n-1}(x)$$ ($a_n >0$ for $n=1,2, \dots$, $b_n$ real, and $p_{-1}(x)\equiv 0$), and orthogonal polynomials on the unit circle (OPUC) where the basic recursion is $${\label}{1.2} 
\varphi_{n+1}(z) = \rho_n^{-1} (z\varphi_n (z) -\bar\alpha_n \varphi_n^*(z))$$ Here $\alpha_n$ are complex coefficients lying in the unit disk ${{\mathbb{D}}}$ and $${\label}{1.3} 
\varphi_n^*(z) = z^n \, 