Simon’s OPUC Hausdorff dimension conjecture

Autor: Darren C. Ong, Shuzheng Guo, David Damanik
Rok vydání: 2021
Předmět:
Zdroj: Mathematische Annalen. 384:1-37
ISSN: 1432-1807
0025-5831
Popis: 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.
Databáze: OpenAIRE
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