$\hat{I}´$-Badly approximable numbers and ubiquitously losing sets

Abstract

We consider a natural filtration $\mathrm{Bad}(\hat{I}´)\subset \mathrm{Bad}(\hat{I}{´}^{\prime })$ for $\hat{I}´\ge \hat{I}{´}^{\prime }>0$ on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the $\ddot{I}„$-well approximable numbers. We show that the set $\mathrm{Bad}(\hat{I}´)$ is a $(1/3,18\hat{I}´)$-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of $(\hat{I}\pm ,{\hat{I}}^2)$-$\text{ubiquitously losing sets}$ to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an $(\hat{I}\pm ,{\hat{I}}^2)$-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that $\mathrm{Bad}(\hat{I}´)$ is a $(1/2,18/\hat{I}´)$-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of $\mathrm{Bad}(\hat{I}´)$ that is strictly less than one. Combined with a finite intersection property and a bilipschitz transfer property, we obtain results for finite intersections of translates of $\mathrm{Bad}(\hat{I}´)$.

77 pages. Introduction revised