Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation
The present research is devoted to a general principle for values of the Hausdorff dimension of the Cartesian product of limsup sets. It is remarked that the metric theory of limsup set is one of the main topics in metric Diophantine approximation.\N\NThe main theorem of this paper describes this principle. Several applications of the main result are presented by examples. The authors explore the following example was never observed before:\N\NSuppose \(W(\psi )\) is the set of \(\psi\)-well approximable points in \(\mathbb R\) and \(\psi : \mathbb N \to \mathbb R^+\) is a positive non-increasing function. Then the relationship \N\[\N\dim_{\mathcal H}W(\psi )\times \cdots \times W(\psi )=d-1+\dim_{\mathcal H}W(\psi )\N\]\Nholds, where \(\dim_{\mathcal H}(\cdot)\) is the Hausdorff dimension of a set.\N\NSpecial attention is also given to the motivation of this research, to a brief survey on related results, and to some generalizations of the main result.
scientific article; zbMATH DE number 7876060
scientific article