The Fine Structure of Dyadically Badly Approximable Numbers

Abstract

We consider badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle $\mathbb{S}$ and the smallest distance to an integer $\|\cdot\|$ we give elementary proofs that the set $F(c) = \{x \in \mathbb{S}: \|2^nx\| \geq c, n\geq 0\}$ is a fractal set whose Hausdorff dimension depends continuously on $c$, is constant on intervals which form a set of Lebesgue measure 1 and is self-similar. Hence it has a fractal graph. Moreover, the dimension of $F(c)$ is zero if and only if $c\geq 1-2\tau$, where $\tau$ is the Thue-Morse constant. We completely characterise the intervals where the dimension remains unchanged. As a consequence we can completely describe the graph of $ c\mapsto \dim_H \{x\in[0,1]: \|x-\frac{m}{2^n}\|< \frac{c}{2^n} \textnormal{finitely often}\}$.