Simon's OPUC Hausdorff Dimension Conjecture

TL;DR

This paper proves Simon's 2005 conjecture that the Hausdorff dimension of the support of the singular part of the measure associated with certain Verblunsky coefficients is bounded by 1 minus the decay rate parameter, using properties of Szeg\

Contribution

It establishes a link between the decay rate of Verblunsky coefficients and the Hausdorff dimension of the measure's support, confirming Simon's conjecture.

Findings

Boundedness of Szeg\

Support of singular measure has Hausdorff dimension ≤ 1 - γ

Confirms Simon's 2005 conjecture

Abstract

We show that the Szeg\H{o} 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.