On the Hausdorff dimension of Weighted Singular Vectors

Abstract

We prove a sharp upper bound on the Hausdorff dimension of weighted singular vectors in $\mathbb{R}^m$ using dynamics on homogeneous spaces, specifically the method of integral inequalities. Together with the lower bound proved recently by Kim and Park \cite{KimPark2024}, this determines the Hausdorff dimension of weighted singular vectors, thereby generalizing to arbitrary dimension, the work of Liao, Shi, Solan, and Tamam \cite{LSST}, who determined the Hausdorff dimension of weighted singular vectors in two dimensions. We also provide the first known bounds for the Hausdorff dimension of weighted singular vectors restricted to fractal subsets.