Effective Hausdorff dimension of bad sets

Abstract: **NOTE THE UNUSUAL TIME**

In this talk, we consider the inhomogeneous Diophantine approximation: the distribution of $qa$ modulo integers near a target real $b$ (for integer $q$ and a real $a$), or more generally $Aq$ modulo integral vectors near a target vector $b$ (where $q$ is an integer vector, and $A$ is a real matrix). We prove that for all $b$, the Hausdorff dimension of the set of matrices that are epsilon badly approximable for the target $b$ is not full, with an effective upper bound. We also give an effective bound on the dimension of the set of targets badly approximated by $Aq$ in terms of epsilon, if the matrix $A$ is not singular on average. The main part of the talk is joint work with Taehyeong Kim and Wooyeon Kim.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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