Abstract
Barry Simon conjectured in 2005 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 γ ∈ (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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Acknowledgements
D.D. was supported in part by NSF grant DMS–1700131 and by an Alexander von Humboldt Foundation research award. J.F. was supported in part by Simons Foundation Collaboration Grant #711663. S.G. was supported by CSC (No. 201906330008) and NSFC (No. 11571327). D.O. was supported in part by two grants from the Fundamental Research Grant Scheme from the Malaysian Ministry of Education (Grant Numbers: FRGS/1/2018/STG06/XMU/02/1 and FRGS/1/2020/STG06/XMU/02/1) and a Xiamen University Malaysia Research Fund (Grant Number: XMUMRF/2020-C5/IMAT/0011).
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Damanik, D., Fillman, J., Guo, S., Ong, D.C. (2021). On Simon’s Hausdorff Dimension Conjecture. In: Gesztesy, F., Martinez-Finkelshtein, A. (eds) From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory . Operator Theory: Advances and Applications, vol 285. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-75425-9_3
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DOI: https://doi.org/10.1007/978-3-030-75425-9_3
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