论文标题
西蒙的Opuc Hausdorff Dimension猜想
Simon's OPUC Hausdorff Dimension Conjecture
论文作者
论文摘要
我们表明，与Verblunsky系数相关的Szegő矩阵$ \ {α_n\} _ {n \ in \ Mathbb {z} _+} $ obeying $ \ sum_ { $ z \ in \ partial \ mathbb {d} $外部一组Hausdorff尺寸不超过$1-γ$。特别是，单位圆上相关概率度量的单数部分由一组Hausdorff尺寸支撑不超过$1-γ$。这证明了Barry Simon从2005年开始的Opuc Hausdorff维度。
We show that the Szegő matrices, associated with Verblunsky coefficients $\{α_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^γ|α_n|^2 < \infty$ for some $γ\in (0,1)$, are bounded for values $z \in \partial \mathbb{D}$ 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.
