On the Hausdorff Dimension of weighted exactly Approximable Vectors

TL;DR

The paper addresses the problem of determining the Hausdorff dimension of the weighted exactly $\tau$-approximable vectors and proves it equals the dimension of the corresponding weighted $\tau$-approximable set for all weights when $\tau>1$. The authors develop a Cantor-set construction $E_{\infty}\subset E_{\boldsymbol w}(\tau)$ guided by an auxiliary weight $\tilde{\boldsymbol w}$ and a four-regime subdivision to accommodate general weights, and they perform a detailed dimension analysis via counting and a mass-distribution argument. A key outcome is an explicit minimization characterization that matches the known dimension formula for $W_{\boldsymbol w}(\tau)$, thereby establishing $\dim_{\mathrm H} E_{\boldsymbol w}(\tau)=\dim_{\mathrm H} W_{\boldsymbol w}(\tau)$. The result generalizes previous equal-weight results, enriching the understanding of metric Diophantine approximation in weighted settings and offering tools for further exploration in weighted approximation problems.

Abstract

We show that the Hausdorff dimension of $\boldsymbol w$-weighted $τ$-exactly approximable vectors in $\mathbb R^d$ coincides with the Hausdorff dimension of $\boldsymbol w$-weighted $τ$-approximable vectors, generalizing a result of the first named author and De Saxcé.

Theorems & Definitions (37)

• Theorem 1: Rynne
• Theorem 2
• Theorem 3: Minkowski’s Second Theorem on Successive Minima
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4