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.