Hausdorff dimension of weighted singular vectors

TL;DR

This paper determines the Hausdorff dimension of weighted singular vectors in two-dimensional space, extending previous results on unweighted vectors by incorporating weights into the dimension calculation.

Contribution

It provides a formula for the Hausdorff dimension of weighted singular vectors in ^2, generalizing prior work on unweighted vectors.

Findings

Hausdorff dimension of weighted singular vectors is 2 - 1/(1 + w_1)

Extends Cheung's results to weighted case

Provides explicit dimension formula depending on weights

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 $\mathbb R^2$.