A general inhomogeneous Jarnik-Besicovitch theorem
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)].
scientific article; zbMATH DE number 1221695
scientific article