Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation
type: Item
instance of: scholarly article
title: Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation
author: Baowei Wang
author: Jun Wu
published in: Transactions of the American Mathematical Society
publication date: 2024-07-04T00:00:00Z
review text: 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.
reviewed by: Symon Serbenyuk
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 7876060
zbMATH Keywords: Cartesian product
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Diophantine approximation
MaRDI profile type: Publication
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DOI: 10.1090/TRAN/9136
description: scientific article; zbMATH DE number 7876060