Hausdorff dimension of weighted singular vectors

Abstract

Let $w=(w_1,w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in ${\mathbb{R}}^2$ has Hausdorff dimension $2-\frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in ${\mathbb{R}}^2$.

References in corpus (2)

• Badly approximable points on manifolds
• Badziahin-Pollington-Velani's theorem and Schmidt's game