Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{ R}$. Let us further assume that $(F_n)_{ n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{ n=1}^\infty$ is a sequence of nonlinear polynomials and $F_n:=f_n\circ\dots\circ f_1$ for all $n\in\mathbb{ N}$. We prove that if there is an $s_0\in\mathbb{ N}$ such that $0$ is a root of $f_n^\prime$ for each $n>s_0$ then the distance between any two zeros of an orthogonal polynomial for $\mu$ of a given degree greater than $1$ has a lower bound in terms of the distance between the set of critical points and the set of zeros of some $F_k$. Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures.
Categories: math.SP math.CA