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This article is cited in 2 scientific papers (total in 2 papers) On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle D. S. Lubinsky School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 11, DOI: https://doi.org/10.4213/sm9569 
Bibliographic databases:
    § 1. IntroductionLet $\mu$ be a finite positive Borel measure on $[-\pi ,\pi)$ (or, equivalently, on the unit circle) with infinitely many points in its support. Then we may define the orthonormal polynomials
$$
\begin{equation*}
\varphi_{n}(z)=\kappa_{n}z^{n}+\dotsb, \qquad \kappa_{n}>0,
\end{equation*}
\notag
$$
$n=0,1,2,\dots$, satisfying the orthonormality conditions
$$
\begin{equation*}
\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi_{n}(z) \overline{\varphi_{m}(z)}\,d\mu (\theta)=\delta_{mn},
\end{equation*}
\notag
$$
where $z=e^{i\theta}$. We denote the zeros of $\varphi_{n}$ by $\{z_{jn}\}_{j=1}^{n}$. They lie inside the unit circle and may not be distinct. Soviet and Russian mathematicians have been leading lights in the theory of orthogonal polynomials ever since Chebyshev laid the foundations. Many in Gonchar’s own school are world leaders, and their students continue that tradition. The celebrated work of Rakhmanov [18], [20], and Aptekarev, Denisov and Tulyakov [3], [4] on Steklov’s conjecture is just one of many examples. It is a privilege to pay tribute to Gonchar’s memory. We shall assume that $\mu$ is regular in the sense of Stahl, Totik and Ullmann [25], so that This is true if, for example, $\mu'>0$ almost everywhere in $[-\pi,\pi)$, but there are pure jump and pure singularly continuous measures that are regular.Many aspects of the zeros $\{z_{jn}\}$ have been studied down the years, for example, their distribution (often when projected onto the unit circle), and ‘clock spacing’ of zeros of paraorthogonal polynomials (see Ch. 8 of Simon’s monograph [23]). One result relevant to this paper is due to Nevai and Totik (see [17] and [23], Theorem 7.1.3). They relate the largest disc centred at the origin and containing all zeros of the orthonormal polynomials to analytic continuation of the Szegő function inside the unit circle. In this case $\mu'$ is infinitely differentiable on the unit circle. Another classic result of Mhaskar and Saff gives sufficient conditions in terms of the recurrence coefficients for the zero counting measures to converge weakly to the uniform distribution on the unit circle (see [15] and [23], Theorem 8.1.2). Breuer and Seelig [7] recently studied clock spacing of zeros of paraorthogonal polynomials, as did Simanek [21], [22] (see also the references there). In a very interesting recent paper Bessonov and Denisov (see [6], Theorem 3) showed that the distance of the zeros to the unit circle is intimately related to the asymptotics of orthogonal polynomials. The following is a reformulation of one of their results. Theorem 1.1. Let $\mu$ be a measure on the unit circle satisfying the Szegő condition For almost every $\zeta$ with $|\zeta|=1$, the following are equivalent: (I)
$$
\begin{equation*}
\lim_{n\to \infty}|\varphi_{n}(\zeta) |^{2}\mu'(\zeta)=1;
\end{equation*}
\notag
$$
(II) We prove related equivalences for local bounds and asymptotics but for regular, rather than Szegő, measures. Theorem 1.2. Let $\mu$ be a finite positive Borel measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\Delta$ be an arc of the unit circle in which $\mu$ is absolutely continuous, while $\mu'$ is positive and continuous there. The following are equivalent: (I) in every proper subarc $\Gamma$ of $\Delta$,
$$
\begin{equation*}
\lim_{n\to \infty}\biggl(\inf \biggl\{n(1-|z_{jn}|)\colon z_{jn}\neq 0,\, \frac{z_{jn}}{|z_{jn}|}\in \Gamma \biggr\}\biggr)=\infty;
\end{equation*}
\notag
$$
(II) in every proper subarc $\Gamma$ of $\Delta$, as $n\to \infty$, uniformly for $z\in \Gamma$, Remarks 1.1. (i) By a proper subarc, we mean that both endpoints of $\Gamma$ are at a positive distance to the endpoints of $\Delta$. All arcs in this paper are assumed to be closed arcs, so contain their endpoints. (ii) We note that if $\mu$ is absolutely continuous on the whole unit circle, while $\mu'$ is positive and continuous there, then by applying the above result to two subarcs, we obtain equivalence on the whole unit circle. As far as the author is aware, even that is new. (iii) The asymptotics of orthogonal polynomials on the unit circle have been studied for at least a century, and there is an extensive literature. If $\log \mu'$ is integrable over the unit circle, then there is an $L_{2}$ asymptotic for $\varphi_{n}$ (see [8], Ch. V, [9], [23], p. 132, and [26], Ch. 10). There are many sufficient conditions for pointwise asymptotics on subarcs of the unit circle, and their real line analogues, and we cannot hope to review these here. The most general result for pointwise asymptotics on the unit circle is almost certainly that of Badkov [5]. He showed that if $\log \mu'$ is integrable on the unit circle and, in some subarc, $\mu$ is absolutely continuous, while $\mu'$ satisfies there a Dini-Lipschitz condition, then we have a uniform asymptotic involving the Szegő function, and hence also (1.1). One of the particularly significant results for non-Szegő weights is due to Rakhmanov (see [19], Theorem 4): if $\mu$ is absolutely continuous on the unit circle, and $\mu'$ satisfies a Dini-Lipschitz condition on the unit circle, then (1.1) holds uniformly on each subarc of the circle where $\mu'$ is bounded below by a positive constant. For bounds, we prove the following. Theorem 1.3. Let $\mu$ be a finite positive Borel measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\Delta$ be an arc of the unit circle in which $\mu $ is absolutely continuous, while $\mu'$ is positive and continuous there. The following are equivalent: (I) in every proper subarc $\Gamma$ of $\Delta$ there exists $C_{1}>0$ such that for $n\geqslant 1$,
$$
\begin{equation*}
\inf \biggl\{n(1-|z_{jn}|) \colon z_{jn}\neq 0,\, \frac{z_{jn}}{|z_{jn}|}\in \Gamma \biggr\} \geqslant C_{1};
\end{equation*}
\notag
$$
(II) in every proper subarc $\Gamma$ of $\Delta$ there exists $C_{2}>0$ such that for $n\geqslant 1$, Remarks 1.2. (i) Again, if $\mu$ is absolutely continuous on the whole unit circle, while $\mu'$ is positive and continuous there, then by applying the above result to two subarcs, we obtain equivalence on the whole unit circle. (ii) Bounds on orthogonal polynomials have also been investigated for a century, with one of the celebrated problems being Steklov’s conjecture. It was Rakhmanov who resolved the conjecture (see [18], [20]), with definitive later contributions by Ambroladze [1], [2], and Aptekarev, Denisov and Tulyakov [3], [4]. There have been many who have contributed in a major way to the broader issue of bounds — for example, Badkov [5], Freud [8], Geronimus [9], Korous and Nevai (see [16]). Again, this is a very incomplete list. (iii) The main ideas underlying the results of this paper are universality limits for reproducing kernels (see [10], [12] and [27]) and local limits for ratios of orthogonal polynomials (see [13]). (iv) For orthogonal polynomials on the real line, the analogous result to Theorem 1.3 involves the distance between zeros of orthogonal polynomials of successive degrees (see [11]). We close this section with more notation. The sin kernel is We let
$$
\begin{equation*}
\varphi_{n}^{\ast}(z)=z^{n}\overline{\varphi_{n}\biggl(\frac{1}{\overline{z}}\biggr)}.
\end{equation*}
\notag
$$
The $n$th reproducing kernel for $\mu$ is
$$
\begin{equation}
K_{n}(z,u)=\sum_{j=0}^{n-1}\varphi_{j}(z) \overline{\varphi_{j}(u)}.
\end{equation}
\tag{1.2}
$$
The Christoffel-Darboux formula asserts that for $z\neq u$,
$$
\begin{equation}
K_{n}(z,u)=\frac{\overline{\varphi_{n}^{\ast}(u)}\varphi_{n}^{\ast}(z) -\overline{\varphi_{n}(u)}\varphi_{n}(z)}{1-\overline{u}z}
\end{equation}
\tag{1.3}
$$
(see [23], p. 954). We let
$$
\begin{equation}
R_{n}(z)=\sum_{j=1}^{n}\frac{1-|z_{jn}|^{2}}{|z-z_{jn}|^{2}}
\end{equation}
\tag{1.4}
$$
and If $z_{jn}=0$, then we set $\tau_{jn}=0$, while if $z_{jn}\neq 0$, then we set Throughout, $C,C_{1},C_{2},\dots$ denote positive constants independent of $n$, $z$, $\zeta$ and polynomials $P$ of degree $\leqslant n$. The same symbol need not denote the same constant in different occurrences. For sequences $\{x_{n}\}$ and $\{y_{n}\}$ of nonzero real numbers we write if there exists $C>1$ independent of $n$, but possibly depending on the sequences, such that
$$
\begin{equation*}
C^{-1}\leqslant \frac{x_{n}}{y_{n}}\leqslant C \quad\text{for all } n\geqslant 1.
\end{equation*}
\notag
$$
This paper is organized as follows: in § 2 we present Theorems 2.1 and 2.2, which state more equivalences than those above. In § 3 we present four preliminary lemmas. We prove Theorem 2.1 in § 4 and Theorem 2.2 in § 5. § 2. Further equivalencesTheorems 1.2 and 1.3 are special cases, respectively, of Theorems 2.1 and 2.2 below. Theorem 2.1. Let $\mu$ be a finite positive Borel measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\Delta$ be an arc of the unit circle in which $\mu$ is absolutely continuous, while $\mu'$ is positive and continuous there. The following are equivalent: (a) uniformly for $z$ in proper subarcs of $\Delta$, (b) uniformly for $z$ in proper subarcs of $\Delta$,
$$
\begin{equation}
\lim_{n\to \infty}\frac{1}{n}\sum_{j=1}^{n}\frac{1-|z_{jn}|^{2}}{|z-z_{jn}|^{2}}=1;
\end{equation}
\tag{2.2}
$$
(c) uniformly for $z$ in proper subarcs of $\Delta$,
$$
\begin{equation}
\lim_{n\to \infty}\operatorname{Re}\biggl(\frac{z\varphi_{n}'(z)}{n\varphi_{n}(z)}\biggr)=1;
\end{equation}
\tag{2.3}
$$
(d) uniformly for $z$ in proper subarcs of $\Delta$,
$$
\begin{equation}
\lim_{n\to \infty}\frac{z\varphi_{n}'(z)}{n\varphi_{n}(z)}=1;
\end{equation}
\tag{2.4}
$$
(e) uniformly for $z$ in proper subarcs of $\Delta$,
$$
\begin{equation}
\lim_{n\to \infty}\frac{\varphi_{n}(ze^{i\pi /n})}{\varphi_{n}(z)}=-1;
\end{equation}
\tag{2.5}
$$
(f) uniformly for $z$ in proper subarcs of $\Delta$ and for $u$ in compact subsets of $\mathbb{C}$,
$$
\begin{equation}
\lim_{n\to \infty}\frac{\varphi_{n}(z(1+u/n))}{\varphi_{n}(z)}=e^{u};
\end{equation}
\tag{2.6}
$$
(g) in proper subarcs $\Gamma$ of $\Delta$,
$$
\begin{equation}
\lim_{n\to \infty}\biggl(\inf \biggl\{n(1-|z_{jn}|) \colon z_{jn}\neq 0,\frac{z_{jn}}{|z_{jn}|}\in \Gamma \biggr\}\biggr)=\infty;
\end{equation}
\tag{2.7}
$$
(h) uniformly for $z$ in proper subarcs of $\Delta$, Remarks 2.1. (i) Weaker versions of parts of Theorem 2.1 appeared in Theorem 1.2 in [13], notably (b), (d), (e) and (f), since we also made an unnecessary assumption (1.7) in [13] about $\operatorname{Im}(\varphi_{n}(ze^{\pm i\pi /n}) /\varphi_{n}(z))$. (ii) There was unfortunately an error in Lemma 4.2, (a), in [13] that led to gaps in proofs below in that paper. These gaps were corrected in [14]. Theorem 2.2. Let $\mu$ be a finite positive Borel measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\Delta$ be an arc of the unit circle in which $\mu$ is absolutely continuous, while $\mu'$ is positive and continuous there. The following are equivalent: (a) for every proper subarc $\Gamma$ of $\Delta$,
$$
\begin{equation}
\sup_{n\geqslant 1}\|\varphi_{n}\|_{L_{\infty}(\Gamma)}<\infty;
\end{equation}
\tag{2.9}
$$
(b) for every proper subarc $\Gamma$ of $\Delta$,
$$
\begin{equation}
\inf_{n\geqslant 1}\inf_{z\in \Gamma}\frac{1}{n} \sum_{j=1}^{n}\frac{1-|z_{jn}|^{2}}{|z-z_{jn}| ^{2}}\geqslant C;
\end{equation}
\tag{2.10}
$$
(c) for every proper subarc $\Gamma$ of $\Delta$, there exists $n_{0}$ such that
$$
\begin{equation}
\inf_{n\geqslant n_{0}}\inf_{z\in \Gamma}\biggl|\operatorname{Re}\biggl(\frac{z\varphi_{n}'(z)}{n\varphi_{n}(z)} -\frac{1}{2}\biggr) \biggr|\geqslant C;
\end{equation}
\tag{2.11}
$$
(d) for every proper subarc $\Gamma$ of $\Delta$, there exists $n_{0}$ such that
$$
\begin{equation}
\inf_{n\geqslant n_{0}}\inf_{z\in \Gamma}\biggl|\operatorname{Re} \biggl(\frac{\varphi_{n}(ze^{\pm i\pi /n})}{\varphi_{n}(z)}\biggr)\biggr|\geqslant C;
\end{equation}
\tag{2.12}
$$
(e) for every proper subarc $\Gamma$ of $\Delta$,
$$
\begin{equation}
\inf \biggl\{n(1-|z_{jn}|) \colon n\geqslant 1,\, z_{jn}\neq 0,\frac{z_{jn}}{|z_{jn}|}\in \Gamma \biggr\} \geqslant C;
\end{equation}
\tag{2.13}
$$
(f) for every proper subarc $\Gamma$ of $\Delta$, § 3. Preliminary lemmasThroughout, we assume the hypotheses of Theorem 1.2, namely, that $\mu$ is regular in the sense of Stahl, Totik and Ullmann, while it is absolutely continuous in $\Delta$, with $\mu'$ positive and continuous there. We first recall some asymptotics for Christoffel functions and universality and local limits. Lemma 3.1. Let $\Gamma$ be a proper subarc of $\Delta$. Then the following hold. (a) Uniformly for $z\in \Gamma$, (b) Uniformly for $z\in \Gamma$ and $a$, $b$ in compact subsets of $\mathbb{C}$,
$$
\begin{equation}
\lim_{n\to \infty}\frac{K_{n}(z(1+i2\pi a/n),z(1+i2\pi \overline{b}/n))}{K_{n}(z,z)} =e^{i\pi (a-b)}\mathbb{S}(a-b).
\end{equation}
\tag{3.2}
$$
(c) Let $\{\zeta_{n}\}\subset \Gamma$. Assume that
$$
\begin{equation}
\sup_{n\geqslant 1}\frac{1}{n}\biggl|\sum_{j=1}^{n}\frac{1}{\zeta_{n}-z_{jn}}\biggr|<\infty \quad\textit{and}\quad\sup_{n\geqslant 1}\frac{1}{n^{2}} \sum_{j=1}^{n}\frac{1}{|\zeta_{n}-z_{jn}|^{2}}<\infty.
\end{equation}
\tag{3.3}
$$
Then from every infinite sequence of positive integers we can choose an infinite subsequence $\mathcal{S}$ such that uniformly for $u$ in compact subsets of $\mathbb{C}$, where Proof. (a) See, for example, [24], Theorem 2.16.1.
(b) See, for example, [10], Theorem 6.3, or [24], Theorem 2.16.1. (c) This follows immediately from Theorem 1.3 in [13] as we have the universality limit (3.2). We note that there was a mistake in Lemma 4.2, (a), in [13] that was corrected in [14]. However, the mistake did not in any way affect the proof of Theorem 1.3 in [13]. Some of the assertions in the following lemma appeared in [13], but we include proofs for the reader’s convenience. Recall that $R_{n}$ and $g_{n}$ were defined, respectively, by (1.4) and (1.5). Lemma 3.2. Let $\Gamma$ be a proper subarc of $\Delta$. Then the following hold. (a) For $|z|=1$,
$$
\begin{equation}
\frac{1}{n}R_{n}(z)=\operatorname{Re}[ 2g_{n}(z) -1].
\end{equation}
\tag{3.6}
$$
(b) Uniformly for $z\in \Gamma$ and fixed real $\alpha$,
$$
\begin{equation}
\lim_{n\to \infty}\operatorname{Im}\bigl[ e^{i\pi \alpha}\varphi_{n}(z) \overline{\varphi_{n}(ze^{2\pi i\alpha /n})}\bigr] \mu'(z)=-\sin \pi \alpha.
\end{equation}
\tag{3.7}
$$
(c) Uniformly for $z\in \Gamma$ and fixed real $\alpha$,
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to \infty}\biggl\{\operatorname{Re}\bigl[ e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(ze^{2\pi i\alpha /n})}\bigr] \frac{1}{n}R_{n}(ze^{2\pi i\alpha /n}) \mu'(z) \\ &\qquad\qquad-(2\sin \pi \alpha) (\operatorname{Im}g_{n}(ze^{2\pi i\alpha /n})) \biggr\}=\cos \pi \alpha. \end{aligned}
\end{equation}
\tag{3.8}
$$
(d) Uniformly for $z\in \Gamma$,
$$
\begin{equation}
\lim_{n\to \infty}\frac{1}{n}R_{n}(z) |\varphi_{n}(z) |^{2}\mu'(z)=1.
\end{equation}
\tag{3.9}
$$
(e) Uniformly for $z\in \Gamma$ and fixed real $\alpha$, Proof. Throughout this proof we assume that with $\alpha$ real or complex.
(a) Elementary manipulations show that
$$
\begin{equation*}
\frac{1-|z_{jn}|^{2}}{|z-z_{jn}|^{2}}=2\operatorname{Re}\biggl(\frac{z}{z-z_{jn}}\biggr) -1.
\end{equation*}
\notag
$$
Dividing by $n$ and adding for $j=1,2,\dots,n$ gives (3.6).
(b) The Christoffel-Darboux formula (1.3) and the universality limit (3.2) (as well as the uniformity of that limit) give uniformly for $\alpha$ in compact subsets of $\mathbb{C}$,
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to \infty}\frac{\overline{\varphi_{n}^{\ast}(z)}\varphi_{n}^{\ast}(\zeta) -\overline{\varphi_{n}(z)}\varphi_{n}(\zeta)}{[ 1-\overline{z}\zeta ] K_{n}(z,z)} =\lim_{n\to \infty}\frac{K_{n}(\zeta,z)}{K_{n}(z,z)} \\ &\qquad =\lim_{n\to \infty}\frac{K_{n}(z(1+\frac{2\pi i\alpha}{n}[ 1+o(1) ]),z)}{K_{n}(z,z)}=e^{i\pi \alpha}\mathbb{S}(\alpha). \end{aligned}
\end{equation}
\tag{3.11}
$$
Here by (3.1),
$$
\begin{equation*}
\lim_{n\to \infty}[ 1-\overline{z}\zeta ] K_{n}(z,z)=-2\pi i\alpha \mu'(z) ^{-1}.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to \infty}[\, \overline{\varphi_{n}^{\ast}(z)}\varphi_{n}^{\ast}(\zeta) -\overline{\varphi _{n}(z)}\varphi_{n}(\zeta) ] \mu'(z) \\ &\qquad =-2\pi i\alpha e^{i\pi \alpha}\mathbb{S}(\alpha) =-2ie^{\pi i\alpha}\sin \pi \alpha=1-e^{2\pi i\alpha}. \end{aligned}
\end{equation}
\tag{3.12}
$$
Next, if $\alpha$ is real, then
$$
\begin{equation*}
\overline{\varphi_{n}^{\ast}(z)}\varphi_{n}^{\ast}(\zeta)=e^{2\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)},
\end{equation*}
\notag
$$
so combining this and (3.12) gives
$$
\begin{equation*}
\lim_{n\to \infty}e^{\pi i\alpha}\bigl\{e^{i\pi \alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)}-e^{-\pi i\alpha} \overline{\varphi_{n}(z)}\varphi_{n}(\zeta) \bigr\} \mu'(z) =-2ie^{\pi i\alpha}\sin \pi\alpha.
\end{equation*}
\notag
$$
Dividing by $2ie^{\pi i\alpha}$ gives (3.7).
(c) We go back to (3.12), which holds uniformly for $\alpha$ in compact subsets of $\mathbb{C}$. This uniformity allows us to differentiate with respect to $\alpha$: after cancelling a factor of $2\pi i$, we obtain
$$
\begin{equation}
\lim_{n\to \infty}\bigl[\, \overline{\varphi_{n}^{\ast}(z)}\varphi_{n}^{\ast \prime}(\zeta) -\overline{\varphi_{n}(z)}\varphi_{n}'(\zeta) \bigr] \frac{\zeta}{n}\mu'(z)=-e^{2\pi i\alpha}.
\end{equation}
\tag{3.13}
$$
Now we again specialize to real $\alpha$, and use that for $|\zeta| =1$,
$$
\begin{equation*}
\varphi_{n}^{\ast \prime}(\zeta)=n\zeta ^{n-1}\overline{\varphi_{n}(\zeta)}-\zeta ^{n-2}\overline{\varphi_{n}'(\zeta)},
\end{equation*}
\notag
$$
so that using $\overline{z}\zeta =e^{2\pi i\alpha/n}$ and recalling the definition (1.5) of $g_{n}$
$$
\begin{equation*}
\overline{\varphi_{n}^{\ast}(z)}\varphi_{n}^{\ast \prime}(\zeta) \frac{\zeta}{n} =e^{2\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)}-e^{2\pi i\alpha} \varphi_{n}(z) \overline{\varphi_{n}(\zeta)}\overline{g_{n}(\zeta)}.
\end{equation*}
\notag
$$
Substituting into (3.13), and dividing by $e^{\pi i\alpha}$,
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to \infty}\bigl[ e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)} -e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)} \overline{g_{n}(\zeta)} \\ &\qquad\qquad-e^{-\pi i\alpha}\overline{\varphi_{n}(z)}\varphi_{n}(\zeta) g_{n} (\zeta) \bigr] \mu'(z) =-e^{\pi i\alpha} \end{aligned}
\end{equation}
\tag{3.14}
$$
or
$$
\begin{equation*}
\lim_{n\to \infty}\bigl[ e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)}-2\operatorname{Re}\{e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)}\overline{g_{n}(\zeta)}\} \bigr] \mu'(z) =-e^{\pi i\alpha}.
\end{equation*}
\notag
$$
Taking real parts,
$$
\begin{equation*}
\lim_{n\to \infty}\operatorname{Re}\bigl[ e^{\pi i\alpha}\varphi _{n}(z) \overline{\varphi_{n}(\zeta)}\{1-2\overline{g_{n}(\zeta)}\} \bigr] \mu'(z)=-\cos \pi \alpha.
\end{equation*}
\notag
$$
Then using (3.7),
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to \infty}\bigl\{\operatorname{Re}\bigl[ e^{\pi i\alpha}\varphi _{n}(z) \overline{\varphi_{n}(\zeta)}\mu'(z) \bigr] \operatorname{Re}[ 1-2g_{n}(\zeta) ] +2\sin \pi \alpha \operatorname{Im}g_{n}(\zeta) \bigr\} \\ &\qquad =-\cos \pi \alpha. \end{aligned}
\end{equation*}
\notag
$$
Finally apply (3.6).
(d) Here we set $\alpha =0$ in (3.8). (e) From (a),
$$
\begin{equation*}
\overline{g_{n}(\zeta)}=2\operatorname{Re}g_{n}(\zeta)-g_{n}(\zeta)=\frac{1}{n}R_{n}(\zeta) +1-g_{n}(\zeta).
\end{equation*}
\notag
$$
We substitute this into (3.14) and cancel a term:
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to \infty}\biggl[-e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)}\frac{1}{n}R_{n}(\zeta ) \\ &\qquad\qquad +g_{n}(\zeta) \bigl\{e^{\pi i\alpha}\varphi_{n}(z) \overline{\varphi_{n}(\zeta)}-e^{-\pi i\alpha}\overline{\varphi_{n}(z)}\varphi_{n}(\zeta) \bigr\} \biggr]\mu'(z) =-e^{\pi i\alpha}. \end{aligned}
\end{equation*}
\notag
$$
Using (3.9) and (3.7), and continuity of $\mu'$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to \infty}\biggl[ -e^{\pi i\alpha}\frac{\varphi_{n}(z)}{\varphi_{n}(\zeta)}(1+o(1) ) +g_{n}(\zeta) 2i(-\sin \pi \alpha) \biggr] =-e^{\pi i\alpha} \\ &\qquad\Longrightarrow\quad \lim_{n\to \infty}\biggl[ \frac{\varphi_{n}(z)}{\varphi_{n}(\zeta)}(1+o(1)) +g_{n}(\zeta) 2ie^{-\pi i\alpha}\sin \pi \alpha \biggr]=1. \end{aligned}
\end{equation*}
\notag
$$
Because of uniformity, we can substitute $ze^{-2\pi i\alpha /n}$ for $z$ so that $\zeta$ becomes $z$.
Lemma 3.2 is proved. We now prove parts of Theorems 2.1 and 2.2. Recall that $\tau_{jn}=z_{jn}/|z_{jn}|$ as in (1.6). Proof. (a), i) $\Rightarrow$ ii). Let $\Gamma$ and $\Gamma_{1}$ be proper subarcs of $\Delta$ such that $\Gamma$ is a proper subarc of $\Gamma_{1}$. In particular, we assume that the distance from both endpoints of $\Gamma$ to the endpoints of $\Gamma_{1}$ is positive. We have for $z\in \Gamma$,
$$
\begin{equation*}
\frac{1}{n^{2}}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1}{|z-z_{jn}|^{2}} \leqslant \frac{1}{Cn}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1-|z_{jn}|^{2}}{|z-z_{jn}|^{2}} \leqslant \frac{1}{Cn}R_{n}(z) \leqslant C_{1}|\varphi_{n}(z) |^{-2}
\end{equation*}
\notag
$$
by (3.9) and positivity and continuity of $\mu'$. Next, we know from (3.7) for $\alpha=1/2$, that
$$
\begin{equation*}
|\varphi_{n}(z) |\,|\varphi_{n}(ze^{i\pi /n}) |\mu'(z)\geqslant 1+o(1),
\end{equation*}
\notag
$$
so it follows that either
$$
\begin{equation*}
\frac{1}{n^{2}}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1}{|z-z_{jn}|^{2}}\leqslant C_{1},
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
\frac{1}{n^{2}}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1}{|ze^{i\pi /n}-z_{jn}|^{2}}\leqslant C_{1}
\end{equation*}
\notag
$$
(or possibly both). But because of our hypothesis, for $\tau_{jn}\in \Gamma_{1}$ and $z\in \Gamma$,
$$
\begin{equation*}
\biggl|\frac{z-z_{jn}}{ze^{i\pi /n}-z_{jn}}\biggr| =\biggl|1+\frac{z(1-e^{i\pi /n})}{ze^{i\pi /n}-z_{jn}}\biggr| \leqslant 1+\frac{2\sin (\pi /(2n))}{1-|z_{jn}|}\leqslant C_{2},
\end{equation*}
\notag
$$
while a similar bound holds for the reciprocal. So for $z\in \Gamma$, Then also for the remaining terms, as the distance from the boundary of $\Gamma$ to that of $\Gamma_{1}$ is positive, and $z\in \Gamma$, Adding the two estimates gives the result.
(a), ii) $\Rightarrow$ i). Choosing $z=\tau_{jn}\in \Gamma$ gives
$$
\begin{equation*}
\frac{1}{n^{2}(1-|z_{jn}|) ^{2}} =\frac{1}{n^{2}|z-z_{jn}|^{2}}\leqslant \frac{1}{n^{2}}\sum_{k=1}^{n}\frac{1}{|z-z_{kn}|^{2}}\leqslant C
\end{equation*}
\notag
$$
by our hypothesis.
(b), i) $\Rightarrow $ ii). Let $\Gamma$ and $\Gamma_{1}$ be as above. For $z\in \Gamma$,
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{n^{2}}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1}{|z-z_{jn}|^{2}} &\leqslant \biggl(\frac{1}{n}\sum_{\tau_{jn}\in\Gamma_{1}}\frac{1-|z_{jn}|^{2}}{|z-z_{jn}|^{2}}\biggr) \frac{1}{\inf_{\tau_{jn}\in \Gamma _{1}}n(1-|z_{jn}|^{2})} \\ &=o\biggl(\frac{1}{n}R_{n}(z)\biggr). \end{aligned}
\end{equation*}
\notag
$$
We can now proceed as in (a).
(b), ii) $\Rightarrow $ i). Again, we proceed much as in (a). Lemma 3.3 is proved. Lemma 3.4. Assume that in every proper subarc $\Gamma$ of $\Delta$, Then in every proper subarc $\Gamma$ of $\Delta$ there exists $C>0$ such that for $n\geqslant 1$, $z_{jn}\neq 0$ and $\tau_{jn}\in \Gamma$, Proof. Let $\Gamma$ be a proper subarc of $\Delta$. Suppose the conclusion is false. Then we can choose an infinite subsequence $\mathcal{S}$ of positive integers and, for $j=j(n)\in \mathcal{S}$, can choose $z_{jn}$ with $\tau_{jn}=z_{jn}/|z_{jn}|\in \Gamma$ such that Write
$$
\begin{equation*}
z_{jn}=\tau_{jn}\biggl(1+2\pi i\frac{\alpha_{n}}{n}\biggr) \quad\text{and}\quad u=\tau_{jn}\biggl(1+2\pi i\frac{\overline{v}}{n}\biggr),
\end{equation*}
\notag
$$
where $v=v(n)$, that is, $v$ depends on $n$ and $\alpha_{n}\to 0$ as $n\to \infty$. Then from the universality limit (3.2) (which by our assumptions holds in a larger arc than $\Gamma$), uniformly for $v$ in compact sets,
$$
\begin{equation}
\frac{K_{n}(z_{jn},u)}{K_{n}(\tau_{jn},\tau_{jn})} =e^{i\pi (\alpha_{n}-v)}\mathbb{S}(\alpha_{n}-v) +o(1) =e^{-i\pi v}\mathbb{S}(v) +o(1).
\end{equation}
\tag{3.15}
$$
Next from the Christoffel-Darboux formula (1.3),
$$
\begin{equation}
\overline{\varphi_{n}^{\ast}(u)}\varphi_{n}^{\ast}(z_{jn}) =\bigl\{K_{n}(\tau_{jn},\tau_{jn}) (1-\overline{u}z_{jn}) \bigr\} \frac{K_{n}(z_{jn},u)}{K_{n}(\tau_{jn},\tau_{jn})},
\end{equation}
\tag{3.16}
$$
and setting $u=\tau_{jn}$, so that $v=0$ in both formulae, and using (3.1), as well as the fact that $n(1-|z_{jn}|)\to 0$, gives
$$
\begin{equation}
\begin{aligned} \, \notag \overline{\varphi_{n}^{\ast}(\tau_{jn})}\varphi_{n}^{\ast}(z_{jn}) &=K_{n}(\tau_{jn},\tau_{jn}) (1-|z_{jn}|) \frac{K_{n}(z_{jn},\tau_{jn})}{K_{n}(\tau_{jn},\tau_{jn})} \\ &=o(1) (1+o(1))=o(1). \end{aligned}
\end{equation}
\tag{3.17}
$$
Now apply (3.15) and (3.16) with $u=\tau_{jn}e^{i\pi /n}$, so that in $u$ above we have ${v=1/2+o(1)}$ and
$$
\begin{equation*}
\frac{K_{n}(z_{jn},u)}{K_{n}(\tau_{jn},\tau_{jn})} =e^{-i\pi /2}\mathbb{S}\biggl(\frac{1}{2}\biggr) +o(1),
\end{equation*}
\notag
$$
while
$$
\begin{equation*}
\begin{aligned} \, &\overline{\varphi_{n}^{\ast}(\tau_{jn}e^{i\pi /n})}\varphi_{n}^{\ast}(z_{jn}) \\ &\qquad =\biggl\{K_{n}(\tau_{jn},\tau_{jn}) \biggl(1-e^{-i\pi /n} \biggl[ 1+2\pi i\frac{\alpha_{n}}{n}\biggr]\biggr) \biggr\} \frac{K_{n}(z_{jn},u)}{K_{n}(\tau_{jn},\tau_{jn})} \\ &\qquad =\biggl\{K_{n}(\tau_{jn},\tau_{jn}) \biggl(1-e^{-i\pi/n} +o\biggl(\frac{1}{n}\biggr)\biggr) \biggr\} \biggl\{e^{-i\pi /2}\mathbb{S}\biggl(\frac{1}{2}\biggr) +o(1) \biggr\}, \end{aligned}
\end{equation*}
\notag
$$
so that using (3.1),
$$
\begin{equation*}
|\varphi_{n}^{\ast}(\tau_{jn}e^{i\pi /n}) \varphi_{n}^{\ast}(z_{jn}) |\sim 1.
\end{equation*}
\notag
$$
Dividing (3.17) by this, gives
$$
\begin{equation*}
\biggl|\frac{\varphi_{n}(\tau_{jn})}{\varphi_{n}(\tau_{jn}e^{i\pi /n})}\biggr| =\biggl|\frac{\varphi_{n}^{\ast}(\tau_{jn})}{\varphi_{n}^{\ast} (\tau_{jn}e^{i\pi /n})}\biggr|=o(1).
\end{equation*}
\notag
$$
But from (3.7) for $\alpha=1/2$, so contradicting the assumed boundedness of $\{\varphi_{n}\}$.
Lemma 3.4 is proved. § 4. Proof of Theorem 2.1We show that assertions (a)–(h) of Theorem 2.1 are equivalent. (a) $\Leftrightarrow $ (b). This is immediate from (3.9). (b) $\Leftrightarrow $ (c). This is immediate from identity (3.6). (c) $\Leftrightarrow $ (d). It is immediate that (2.4) $\Rightarrow$ (2.3). Now assume that (2.3) holds. We must show that $\operatorname{Im}g_{n}(z)\to 0$ as $n\to \infty$, uniformly in $\Gamma$. From the established equivalence (b) $\Leftrightarrow$ (c) we have (2.2), so from (3.8) with $\alpha=1/2$, and (3.9),
$$
\begin{equation}
\lim_{n\to \infty}\bigl\{-\operatorname{Im} [ \varphi_{n}(z) \overline{\varphi_{n}(ze^{i\pi /n})}\mu'(z) ] -2(\operatorname{Im}g_{n}(ze^{\pi i/n})) \bigr\}=0.
\end{equation}
\tag{4.1}
$$
Next, we have already proved that (c) is equivalent to (a), so using (2.1) for ${\zeta =z,ze^{i\pi /n}}$ and continuity of $\mu'$ we have
$$
\begin{equation*}
\lim_{n\to \infty}|\varphi_{n}(z) \overline{\varphi_{n}(ze^{i\pi /n})}|\mu'(z)=1,
\end{equation*}
\notag
$$
while from (3.7) for $\alpha=1/2$,
$$
\begin{equation*}
\lim_{n\to \infty}\operatorname{Re}\bigl[ \varphi_{n}(z) \overline{\varphi_{n}(ze^{i\pi /n})}\bigr] \mu'(z)=-1.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\lim_{n\to \infty}\operatorname{Im}\bigl[ \varphi_{n}(z) \overline{\varphi_{n}(ze^{i\pi /n})}\bigr] \mu'(z)=0.
\end{equation*}
\notag
$$
Then (4.1) gives
$$
\begin{equation}
\lim_{n\to \infty}\operatorname{Im}g_{n}(ze^{\pi i/n})=0.
\end{equation}
\tag{4.2}
$$
All of the above limits hold uniformly in $\Gamma$, and even in a larger subarc of $\Delta$. Because of the uniformity in $z$ we may replace $ze^{i\pi /n}$ by $z$. So indeed (2.3) $\Rightarrow $ (2.4). (d) $\Leftrightarrow $ (e). From (3.10) with $\alpha=-1/2$,
$$
\begin{equation*}
2g_{n}(z)=1-\frac{\varphi_{n}(ze^{\pi i/n})}{\varphi_{n}(z)}(1+o(1)) +o(1),
\end{equation*}
\notag
$$
and so (2.4) $\Leftrightarrow $ (2.5). (a) $\Leftrightarrow $ (f). Let $\Gamma_{1}$ be a proper subarc of $\Delta$ properly containing $\Gamma$. First assume that (2.1) holds. We apply Lemma 3.1, (c), for all $\zeta_{n}=z$, and must verify (3.3) there. The first condition in (3.3) for all $\zeta_{n}=z$ follows immediately from (2.4) — and in turn, we have proved that follows from (2.1). For the second, first observe from Lemma 3.4 and our hypothesis that
$$
\begin{equation*}
\inf \{n(1-|z_{jn}|) \colon \tau_{jn}\in\Gamma \} \geqslant C.
\end{equation*}
\notag
$$
Then Lemma 3.3, (a), shows that the second condition in (3.3) holds for all $\zeta_{n}=z$. From Lemma 3.1, (c), we obtain that every subsequence of positive integers contains a further subsequence $\mathcal{S}$ such that uniformly for $u$ in compact subsets of $\mathbb{C}$,
$$
\begin{equation*}
\lim_{n\to \infty,\, n\in \mathcal{S}}\frac{\varphi_{n}(z(1+u/n))}{\varphi_{n}(z)}=e^{u}+C(e^{u}-1),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
C=\lim_{n\to \infty,\,n\in \mathcal{S}} \biggl(\frac{z}{n}\frac{\varphi_{n}'(z)}{\varphi_{n}(z)}-1\biggr).
\end{equation*}
\notag
$$
But from (2.4) (which, as we know, follows from (a)) $C=0$, so the limit is independent of the subsequence, and we have (2.6). Now conversely assume that we have the local limit (2.6). Then setting $u=i\pi /n$ and using the uniformity,
$$
\begin{equation*}
\lim_{n\to \infty}\frac{\varphi_{n}(ze^{i\pi /n})}{\varphi_{n}(z)} =\lim_{n\to \infty}\frac{\varphi_{n}(z(1+\frac{i\pi}{n}[ 1+o(1) ]))}{\varphi_{n}(z)} =e^{i\pi}=-1,
\end{equation*}
\notag
$$
so we have (2.5) and hence the required result from the established equivalence (a) $\Leftrightarrow $ (e). (f) $\Rightarrow $ (g). This is a consequence of the fact that $e^{u}$ has no zeros. Indeed, if there were a subsequence of zeros $z_{jn}$, $n\in \mathcal{S}$, $j=j(n)$, with $\tau_{jn}\in \Gamma$ and $1-|z_{jn}|=O(1/n)$, then writing $z_{jn}=\tau_{jn}(1+i\alpha_{n}/n)$, we have $\alpha_{n}=O(1)$, and by the local limit,
$$
\begin{equation*}
0=\frac{\varphi_{n}(\tau_{jn}(1+i\alpha_{n}/n))}{\varphi_{n}(\tau_{jn})} =e^{\pi i\alpha_{n}}+o(1),
\end{equation*}
\notag
$$
leading to a contradiction. (g) $\Leftrightarrow $ (h). This is Lemma 3.3, (b). (h) $\Rightarrow $ (f). Now,
$$
\begin{equation*}
\frac{1}{n}g_{n}'(z) =\frac{1}{n^{2}}\frac{d}{dz} \biggl(\sum_{j=1}^{n}\biggl[ 1+\frac{z_{jn}}{z-z_{jn}}\biggr]\biggr) =-\frac{1}{n^{2}}\sum_{j=1}^{n}\frac{z_{jn}}{(z-z_{jn}) ^{2}}.
\end{equation*}
\notag
$$
Let $A>0$. Our hypothesis gives uniformly for $z\in \Gamma$, and hence for $\zeta,z\in\Gamma$ with $|\zeta-z|\leqslant A/n$, Next (3.10) for $\alpha=-1/2$ gives
$$
\begin{equation*}
2g_{n}(z)=1-\frac{\varphi_{n}(ze^{i\pi /n})}{\varphi_{n}(z)}(1+o(1)) +o(1).
\end{equation*}
\notag
$$
Also, replacing $z$ by $ze^{i\pi /n}$ and using (3.10) for $\alpha=1/2$ gives
$$
\begin{equation*}
2g_{n}(ze^{i\pi /n})=1-\frac{\varphi_{n}(z)}{\varphi_{n}(ze^{i\pi /n})}(1+o(1)) +o(1).
\end{equation*}
\notag
$$
Thus from (4.3),
$$
\begin{equation*}
\biggl|\frac{\varphi_{n}(ze^{i\pi /n})}{\varphi_{n}(z)}(1+o(1)) -\frac{\varphi_{n}(z)}{\varphi_{n}(ze^{i\pi /n})}(1+o(1)) \biggr|=o(1),
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\biggl(\frac{\varphi_{n}(ze^{i\pi /n})}{\varphi_{n}(z)}\biggr)^{2}=1+o(1).
\end{equation*}
\notag
$$
In view of (3.7) for $\alpha=1/2$, necessarily
$$
\begin{equation*}
\lim_{n\to \infty}\frac{\varphi_{n}(ze^{i\pi /n})}{\varphi_{n}(z)}=-1.
\end{equation*}
\notag
$$
Then we have the conclusion (2.5), and the established equivalence (e) $\Leftrightarrow $ (f) gives the result. Note that all the limits above hold uniformly in $\Gamma$, so we have uniformity in (2.5). Theorem 2.1 is proved. § 5. Proof of Theorem 2.2We note that several of the equivalences hold automatically for finitely many $n$. So we should deal with large enough $n$. (a) $\Leftrightarrow $ (b). This is immediate from (3.9) and the continuity of $\mu'$. (b) $\Leftrightarrow $ (c). This is immediate from (3.6). (c) $\Leftrightarrow $ (d). This follows from (3.10) with $\alpha=-1/2$, which implies that
$$
\begin{equation*}
2g_{n}(z) -1=-\frac{\varphi_{n}(ze^{\pm i\pi /n})}{\varphi_{n}(z)}(1+o(1)) +o(1).
\end{equation*}
\notag
$$
(a) $\Rightarrow $ (e). This was proved in Lemma 3.4. (e) $\Leftrightarrow $ (f). This was proved in Lemma 3.3, (a). (f) $\Rightarrow $ (a). Assume the result is false. Then we can choose $\Gamma\subset\Delta$, a sequence $\mathcal{S}$ of positive integers, and for $n\in \mathcal{S}$ we can choose $\zeta_{n}\in \Gamma$ such that Then from (3.9), Let $C>0$. Let $\{u_{n}\}$ be a sequence on the unit circle such that $|\zeta_{n}-u_{n}|\leqslant C/n$, $n\geqslant1$. We claim that Indeed, let $\Gamma_{1}$ contain $\Gamma$ as a proper subarc. If $z_{jn}\in \Gamma_{1}$, then using (2.13) (which we may because of the equivalence (e) $\Leftrightarrow $ (f))
$$
\begin{equation*}
\biggl|\frac{u_{n}-z_{jn}}{\zeta_{n}-z_{jn}}\biggr|\leqslant 1+\frac{|u_{n}-\zeta_{n}|}{1-|z_{jn}|}\leqslant C
\end{equation*}
\notag
$$
with a similar lower bound. The terms containing $z_{jn}\notin \Gamma_{1}$ are easier, so $R_{n}(\zeta_{n})\sim R_{n}(u_{n})$ and we have (5.1). Next, using the equivalence (e) $\Leftrightarrow$ (f),
$$
\begin{equation*}
\frac{1}{n^{2}}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1}{|u_{n}-z_{jn}|^{2}} \leqslant \frac{C}{n}\sum_{\tau_{jn}\in \Gamma_{1}}\frac{1-|z_{jn}|^{2}}{|u_{n}-z_{jn}|^{2}} \leqslant\frac{C}{n}R_{n}(u_{n})=o(1),
\end{equation*}
\notag
$$
while the tail sum admits the estimate
$$
\begin{equation*}
\frac{1}{n^{2}}\sum_{\tau_{jn}\notin \Gamma_{1}}\frac{1}{|u_{n}-z_{jn}|^{2}}\leqslant C\frac{n}{n^{2}}=o(1),
\end{equation*}
\notag
$$
so
$$
\begin{equation*}
\frac{1}{n^{2}}\sum_{j=1}^{n}\frac{1}{|u_{n}-z_{jn}|^{2}}=o(1).
\end{equation*}
\notag
$$
We now proceed much as in the proof of (h) $\Rightarrow $ (f) in Theorem 2.1. We have
$$
\begin{equation*}
\frac{1}{n}g_{n}'(u_{n})=-\frac{1}{n^{2}}\sum_{j=1}^{n}\frac{z_{jn}}{(u_{n}-z_{jn}) ^{2}}=o(1).
\end{equation*}
\notag
$$
It follows that if $|u_{n}-\zeta_{n}|\leqslant C/n$, and $u_{n},\zeta_{n}\in\Gamma$, then Then from (3.10) for $\alpha=1/2$ and appropriate choices of $u_{n}$,
$$
\begin{equation*}
\begin{aligned} \, &\biggl|\frac{\varphi_{n}(\zeta_{n}e^{i\pi /n})}{\varphi_{n}(\zeta_{n})}(1+o(1)) -\frac{\varphi_{n}(\zeta_{n})}{\varphi_{n}(\zeta_{n}e^{i\pi/n})}(1+o(1)) \biggr|=o(1) \\ &\qquad \Longrightarrow\quad \biggl(\frac{\varphi_{n}(\zeta_{n}e^{i\pi /n})}{\varphi_{n}(\zeta_{n})}\biggr) ^{2}=1+o(1) \\ &\qquad \Longrightarrow\quad\frac{\varphi_{n}(\zeta_{n}e^{i\pi /n})}{\varphi_{n}(\zeta_{n})}=-1+o(1) \end{aligned}
\end{equation*}
\notag
$$
in view of (3.7) for $\alpha=1/2$. Next, using (3.10), which gives using (3.6) that and hence using (3.9), that This contradicts our assumption that $\{\varphi_{n}(\zeta_{n})\}$ is unbounded. Theorem 2.2 is proved. 
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