On Simon's Hausdorff Dimension Conjecture

Simon's OPUC Hausdorff Dimension Conjecture

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.

Large Deviations and the Lukic Conjecture

We use the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault to prove higher order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two singular points, one simple and one double. This is important because it is known that the conjecture of Simon fails in exactly this case, so this paper provides support for the idea that Lukic's replacement for Simon's conjecture might be true.

高等Szegő定理中Simon-Lukic猜想的完整证明

A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szegő Theorems

本文通过结合单位变换、离散Sobolev不等式、高阶Szegő展开和新代数分解技术，证明了单位圆上正交多项式中Simon-Lukic猜想的完整性。

临界几乎Mathieu算子：隐藏奇异性、谱隙连续性与谱的Hausdorff维数

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

本文通过将临界几乎Mathieu族表示为奇异Jacobi矩阵，证明了其谱的Hausdorff维数不超过1/2，解决了长期问题，并推广了谱测度连续性与量子图石墨烯的维数结果。

On a conjecture for higher-order Szego theorems

We disprove a conjecture of Simon for higher-order Szego theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture.