On Simon's Hausdorff Dimension Conjecture

Abstract

Barry Simon conjectured in 2005 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 $1-\gamma$. Three of the authors recently proved this conjecture by employing a Prüfer variable approach that is analogous to work Christian Remling did on Schrödinger operators. This paper is a companion piece that presents a simple proof of a weak version of Simon's conjecture that is in the spirit of a proof of a different conjecture of Simon.

9 pages