Date | 2017-10-27 |
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Speaker | Mumtaz Hussain |

Dept. | La Trobe University |

Room | 129-104 |

Time | 15:00-17:00 |

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem, that is if

the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$

has a non-trivial integer solution for all large enough $t$. In this talk, I will explain that the Hausdorff measure of the set of $\psi$-Dirichlet non-improvable numbers obeys a zero-infinity law for a large class of dimension functions.

Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

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