Simon's OPUC Hausdorff Dimension Conjecture

Abstract

We show that the Szegő matrices, associated with Verblunsky coefficients $\{{\alpha }_n{\}}_{n\in {\mathbb{Z}}_{+}}$ obeying ${\sum }_{n=0}^{\infty }{n}^{\gamma }|{\alpha }_n{|}^2<\infty$ for some $\gamma \in (0,1)$, are bounded for values $z\in \partial \mathbb{D}$ outside a set of Hausdorff dimension no more than $1-\gamma$. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than $1-\gamma$. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.

33 pages