# On the spacing of zeros of paraorthogonal polynomials for singular measures

## TL;DR

This paper establishes lower bounds on the spacing of zeros of paraorthogonal polynomials on the unit circle, linking measure continuity with zero distribution, and demonstrates clock spacing for some singular continuous measures.

## Contribution

It provides new bounds on zero spacing based on measure continuity and extends clock spacing results to certain singular continuous measures.

## Key findings

- Lower bounds on zero spacing based on Hausdorff dimensions
- Clock spacing holds for some singular continuous measures
- Extension of spacing results to singular measures

## Abstract

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from arXiv:1011.3159 showing that clock spacing holds even for certain singular continuous measures.