A general inhomogeneous Jarnik-Besicovitch theorem
type: Item
instance of: scholarly article
title: A general inhomogeneous Jarnik-Besicovitch theorem
zbMATH Open document ID: 0917.11036
published in: Journal of Number Theory
publication date: 1999-07-20T00:00:00Z
review text: Let \({\mathbf\beta}\in{\mathbb R}^n\) be some fixed vector, \(\psi:{\mathbb N}\to{\mathbb R}^{+}\) a decreasing positive real-valued function converging to \(0\) at \(\infty\) and let \({\mathbf q}=(q_1,\dots,q_m)\) be a non-zero element of \({\mathbb Z}^m\). Let \(W(\psi,{\mathbf\beta})\) be the set of \(X\) in \({\mathbb R}^{mn}\) such that \(| \langle q_1 x_{1j}+\cdots+q_m x_{mj}-\beta_j\rangle| <\psi(| {\mathbf q}|)\) for \(j=1,\dots,n\), or \(| \langle{\mathbf q}X-{\mathbf\beta}\rangle| <\psi(| {\mathbf q}|)\), for infinitely many \(\mathbf q\) in \({\mathbb Z}^m\backslash\{{\mathbf 0}\}\). Here, \(\langle x\rangle=\{x\}\) if \(\{x\}\leq 1/2\); \(\{x\}-1\) if \(\{x\}>1/2\), where \(\{x\}\) is the fractional part of \(x\). In this paper an inhomogeneous `singly' metric version of the generalization of the Jarnik-Besicovitch theorem to systems of linear forms is established. The Hausdorff dimension of the set \(W(\psi,{\mathbf\beta})\) is determined. A `doubly' metric counterpart of \(W(\psi,{\mathbf\beta})\), where \(\mathbf\beta\) is no longer a fixed vector, is studied by \textit{H. Dickinson} [Glasg. Math. J. 39, 233-236 (1997; Zbl 0882.11045)] and \textit{M. Dodson} [J. Aust. Math. Soc., Ser. A 62, 175-185 (1997; Zbl 0887.11031)].
reviewed by: Takao Komatsu
Mathematics Subject Classification ID: 11K60
Mathematics Subject Classification ID: 11J20
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 1221695
zbMATH Keywords: inhomogeneous diophantine approximation
zbMATH Keywords: Jarnik-Besicovitch theorem
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: resonant set
zbMATH Keywords: ubiquitous system
author: Jason Levesley
MaRDI profile type: Publication
full work available at URL: jnth.1998.2239
OpenAlex ID: W2139064479
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DOI: 10.1006/JNTH.1998.2239
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description: scientific article; zbMATH DE number 1221695