Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation

Abstract

The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that dim H ⁡ W ( ψ ) × ⋯ × W ( ψ ) = d − 1 + dim H ⁡ W ( ψ ) \begin{equation*} \dim _{\mathcal H}W(\psi )\times \cdots \times W(\psi )=d-1+\dim _{\mathcal H}W(\psi ) \end{equation*} where W ( ψ ) W(\psi ) is the set of ψ \psi -well approximable points in R \mathbb {R} and ψ : N → R + \psi : \mathbb {N}\to \mathbb {R}^+ is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand’s famous inequality on the dimension of the Cartesian product of general sets.