$\psi$-badly approximable points

In this talk I will discuss recent work with Henna Koivusalo, Jason Levesley, and Xintian Zhang on the set of $\psi$-badly approximable points. $\psi$-badly approximable points are those which are $\psi$-well approximable, but at the same time not $c\psi$-well approximable for arbitrary small constant $c>0$. In 2003 Bugeaud proved in the one dimensional setting that the Hausdorff dimension of $\psi$-badly approximable points is the same as the Hausdorff dimension of $\psi$-well approximable points. Our main result provides a partial $d$-dimensional analogue of Bugeaud'sresult. In order to do this we construct a Cantorset that simultaneously captures the well approximable and badly approximable nature of $\psi$-badly approximable points.

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