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arXiv · 2012.00660

On Simon's Hausdorff Dimension Conjecture

David Damanik·Jake Fillman·Shuzheng Guo·Darren C. Ong
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Abstract

Barry Simon conjectured in 2005 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 $1 - γ$. Three of the authors recently proved this conjecture by employing a Prüfer variable approach that is analogous to work Christian Remling did on Schrödinger operators. This paper is a companion piece that presents a simple proof of a weak version of Simon's conjecture that is in the spirit of a proof of a different conjecture of Simon.

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David Damanik, Jake Fillman, Shuzheng Guo, Darren C. Ong. 2020-12-01. On Simon's Hausdorff Dimension Conjecture. https://arxiv.org/abs/2012.00660

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