Abstract
We show 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 \({\gamma } \in (0,1)\), are bounded for values \(z \in \partial {{\mathbb {D}}}\) outside a set of Hausdorff dimension no more than \(1 - {\gamma }\). 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 \(1-{\gamma }\). This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.
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References
Christ, M., Kiselev, A.: WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials. Commun. Math. Phys. 218, 245–262 (2001)
Damanik, D.: Verblunsky coefficients with Coulomb-type decay. J. Approx. Theory 139, 257–268 (2006)
Damanik, D., Killip, R.: Half-line Schrödinger operators with no bound states. Acta Math. 193, 31–72 (2004)
Denisov, S., Kupin, S.: On the singular spectrum of Schrödinger operators with decaying potential. Trans. Am. Math. Soc. 357, 1525–1544 (2005)
Falconer, K.: Techniques in Fractal Geometry. Wiley, Chichester (1997)
Golinskiĭ, B., Ibragimov, I.: A limit theorm of G. Szegő. Izv. Akad. Nauk SSSR Ser. Mat. 35, 408–427 (1971)
Kiselev, A.: An interpolation theorem related to the a.e. convergence of integral operators. Proc. Am. Math. Soc. 127, 1781–1788 (1999)
Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Commun. Math. Phys. 193, 151–170 (1998)
Remling, C.: Bounds on embedded singular spectrum for one-dimensional Schrödinger operators. Proc. Am. Math. Soc. 128, 161–171 (2000)
Simon, B.: The Golinskii–Ibragimov method and a theorem of Damanik and Killip. Int. Math. Res. Not. 36, 1973–1986 (2003)
Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. AMS Colloquium Publications, Part 1, vol. 54. American Mathematical Society, Providence (2005)
Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. AMS Colloquium Publications, Part 2, vol. 54. American Mathematical Society, Providence (2005)
Simon, B.: Szegő’s Theorem and Its Descendants. Spectral Theory for \(L^2\) Perturbations of Orthogonal Polynomials. M. B. Porter Lectures. Princeton University Press, Princeton (2011)
Funding
D.O. was supported in part by a grant from the Fundamental Research Grant Scheme from the Malaysian Ministry of Education (Grant no. FRGS/1/2018/STG06/XMU/02/1) and a Xiamen University Malaysia Research Fund (Grant Number: XMUMRF/2020-C5/IMAT/0011). D.D. was supported in part by Simons Fellowship \(\# 669836\), NSF grants DMS-1700131 and DMS-2054752, and an Alexander von Humboldt Foundation research award. S.G. was supported by NSFC (no. 11571327) and CSC (no. 201906330008).
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Communicated by Loukas Grafakos.
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Damanik, D., Guo, S. & Ong, D.C. Simon’s OPUC Hausdorff dimension conjecture. Math. Ann. 384, 1–37 (2022). https://doi.org/10.1007/s00208-021-02283-7
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DOI: https://doi.org/10.1007/s00208-021-02283-7