Extravagance, irrationality and Diophantine approximation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911137587527680 |
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| author | Aaronson, Jon Nakada, Hitoshi |
| author_facet | Aaronson, Jon Nakada, Hitoshi |
| contents | For an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotient function is integrable. We show that with respect to a ``continued fraction mixing'' measure for the Gauss map with the log of the partial quotient function non-integrable, almost all numbers are Liouville. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are applications of our study of the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type dichotomy for Diophantine approximation with respect to ergodic``weak Renyi measures'' which are ``doubling at $0$''. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_19393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extravagance, irrationality and Diophantine approximation Aaronson, Jon Nakada, Hitoshi Dynamical Systems Number Theory Probability 11K50, 37A44, 60F20 For an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotient function is integrable. We show that with respect to a ``continued fraction mixing'' measure for the Gauss map with the log of the partial quotient function non-integrable, almost all numbers are Liouville. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are applications of our study of the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type dichotomy for Diophantine approximation with respect to ergodic``weak Renyi measures'' which are ``doubling at $0$''. |
| title | Extravagance, irrationality and Diophantine approximation |
| topic | Dynamical Systems Number Theory Probability 11K50, 37A44, 60F20 |
| url | https://arxiv.org/abs/2409.19393 |