The Hausdorff dimension of sets of points whose simultaneous rational approximation by sequences of integer vectors have errors with small product (Q1333251)

Restricted diophantine approximation in which the integer vectors are drawn from an infinite subset \(Q\) of \(\mathbb{Z}^ n\) is considered. Let \[ E_ Q= \Biggl\{X\in \mathbb{R}^{mn}:\;\prod_{i=1}^ m\;\Biggl\| \sum_{j=1}^ n x_{ij} q_ j \Biggr\|< |{\mathbf q}|^{- \tau} \text{ for infinitely many }{\mathbf q}\in Q\Biggr\}, \] where \({\mathbf q}= (q_ 1,\dots, q_ n)\) and \(| {\mathbf q}|= \max\{| q_ 1|,\dots,| q_ n|\}\). Then the Hausdorff dimension of \(E_ Q\) is \(mn-1+ (\nu+1)/ (\tau+1)\), where \(\nu\in [0,n]\) is the exponent of convergence for the series \(\sum_{{\mathbf q}\in Q} |{\mathbf q}|^{-s}\).