Simon’s OPUC Hausdorff dimension conjecture

We show that the Szegő matrices, associated with Verblunsky coefficients { α n } n ∈ Z + obeying ∑ n = 0 ∞ n γ | α n | 2 < ∞ for some γ ∈ ( 0 , 1 ) , are bounded for values z ∈ ∂ D outside a set of Hausdorff dimension no more than 1 - γ . 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 - γ . This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.