On a lower bound of Hausdorff dimension of weighted singular vectors

TL;DR

This paper establishes a new lower bound for the Hausdorff dimension of the set of weighted singular vectors in Euclidean space, extending previous results to higher dimensions and general weight configurations.

Contribution

It provides a novel lower bound for the Hausdorff dimension of weighted singular vectors, generalizing earlier work from two dimensions to arbitrary dimensions with weights.

Findings

Lower bound of Hausdorff dimension is at least d - 1/(1+w_1)

Extends previous 2D results to higher dimensions

Advances understanding of weighted Diophantine approximation sets

Abstract

Let $w=(w_1,\dots,w_d)$ be a $d$-tuple of positive real numbers such that $\sum_{i}w_i =1$ and $w_1\geq \cdots \geq w_d$. A $d$-dimensional vector $x=(x_1,\dots,x_d)\in\mathbb{R}^d$ is said to be $w$-singular if for every $\epsilon>0$ there exists $T_0>1$ such that for all $T>T_0$ the system of inequalities \[ \max_{1\leq i\leq d}|qx_i - p_i|^{\frac{1}{w_i}} < \frac{\epsilon}{T} \quad\text{and}\quad 0<q<T \] have an integer solution $(\mathbf{p},q)=(p_1,\dots,p_d,q)\in \mathbb{Z}^d \times \mathbb{Z}$. We prove that the Hausdorff dimension of the set of $w$-singular vectors in $\mathbb{R}^d$ is bounded below by $d-\frac{1}{1+w_1}$. Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in $\mathbb{R}^2$, J. Eur. Math. Soc. 22 (2020), 833-875].