holds, we can view the $a$’s and $b$’s as periodic of period $mp$. There are closed gaps where $T_{mp}(z)=\pm\boldsymbol{1}$, that is, interior points to the original bands where $(\Gamma_\pm)^m =1$, that is, points where holds. Thus, the Dirichlet data for $T_{mp}$ are exactly the points claimed.

Theorem \[T2.1\] immediately implies point (2) from the introduction.

[T2.3]{} Let $P_n(x)$ be a family of OPRL associated to a set of Jacobi parameters obeying . Let $(\alpha_j,\beta_j)$ be a single band and let $N^{(n,j)}$ be the number of zeros of $P_n$ in that band. Then $${\label}{2.20} 
{\lvertN^{(mp+b,j)} - (m-1)\rvert} \leq \min (b+1, p-b)$$ for $-1\leq b\leq p-1$. In particular, $${\label}{2.21} 
\biggl| N^{(n,j)} - {\frac}{n}{p}\biggr| \leq 1 + {\frac}{p}{2}$$

By a variational principle for any $n,n'$, $${\label}{2.22} 
{\lvertN^{(n,j)} - N^{(n',j)}\rvert} \leq {\lvertn-n'\rvert}$$ is immediate from Theorem \[T2.1\] if we take $n'=mp-1$ and $n'=
mp+(p-1)$. follows from given that $\min (b+1,p-b) 
\leq p/2$.

[*Remark.*]{} Because of possibilities of Dirichlet data zeros at $\alpha_j$ and/or $\beta_j$, we need $(\alpha_j, \beta_j)$ in defining $N^{(n,j)}$. It is more natural to use $[\alpha_j, \beta_j]$. If one does that, becomes $2+\min(b+1,p-b)$ and , $3+{\frac}{p}{2}$.

To go beyond these results and prove clock behavior for the zeros of $p_{mp+b}$ ($b\not\equiv -1$ 