The Hausdorff dimension of sets arising from diophantine approximation with a general error function
Let \(m,n\) be any positive integers and let \(\psi\) be a bounded non-negative function without monotonicity conditions. The author shows that the subset \(W(m,n;\psi)\) of the set \(M_{m\times n}(\mathbb{R})\) (identified with \(\mathbb{R}^{mn})\) of \(m\times n\) real matrices \((x_{ij})\) which satisfy the system of inequalities \[ \left\|\sum^n_{j=1}x_{ij}q_j \right\| <\psi({\mathbf q}), \quad 1\leq i\leq m, \] for infinitely many \({\mathbf q}=(q_1, \dots,q_n)\in \mathbb{Z}^n\), has Hausdorff dimension \[ \dim W(m,n; \psi)=m(n-1)+\min \bigl\{ \eta (\psi),m\bigr\}=m(n-1)+ \min\bigl\{\delta (\psi),m\bigr\}, \] where \(\eta (\psi)\) is the exponent of convergence of the sum \(\sum_{0\neq{\mathbf q}\in \mathbb{Z}^n} |{\mathbf q}|^m\bigl(\psi ({\mathbf q})/ |{\mathbf q}| \bigr)^\eta\), and \(\delta (\psi)\) is related to the order of \(\psi\). This is a very nice general result which reduces the question of dimension to determining \(\eta(\psi)\) or \(\delta(\psi)\).
scientific article; zbMATH DE number 1236178