For given \(s\in {\mathbb{N}}\), let \(\theta_ s\) denote the supremum of all reals c with the property that, for any vector \({\bar \alpha}=(\alpha_ 1,...,\alpha_ s)\in({\mathbb{R}}^ s-{\mathbb{Q}}^ s),\) there exist infinitely many \((\bar p,q)\in {\mathbb{Z}}^ s\times {\mathbb{N}}\) satisfying \(| {\bar \alpha}-(1/q)\bar p| \leq c^{-1/s}\quad q^{-1-1/s}\) where \(| . |\) denotes the Euclidean norm. (The values \(\theta_ 1=\sqrt{5}\) and \(\theta_ 2=\sqrt{23}/2\) are known due to Hurwitz (classic) and \textit{H. Davenport} and \textit{K. Mahler} [Duke Math. J. 13, 105-111 (1946; Zbl 0060.120)].) In this note a method of \textit{H. Davenport} [J. Lond. Math. Soc. 13, 139-145 (1938; Zbl 0018.29503)] and \textit{G. Žilinskas} [J. Lond. Math. Soc. 16, 27-37 (1941; Zbl 0028.34901)] is applied to derive new lower bounds for \(\theta_ s\), \(s\geq 4\) and results in the Theorem: \[ \theta_ s\geq(s+1)^{(s+1)/2}\quad s^{-s/2}\quad \Delta(S_{s+1})k_ s, \] where \(k_ s=(\min \{z(Q_ 1),\quad z(Q_ 2)\})^{s/2}<1, z(Q)=(Q^{-s}+sQ)/(s+1),\) the \(Q_ i (i=1,2)\) are the (unique) positive zeros of \(L_ i(Q)=Q^{-s}+(-1)^ i(1+Q)^{-s}-1\) and \(\Delta(S_{s+1})\) is the critical determinant of the \((s+1)\)- dimensional unit sphere. Using the known values of \(\Delta(S_{s+1})\) for \(s\leq 7\), one obtains the numerical results \(\theta_ 4\geq 1,23645,\quad \theta_ 5\geq 0,83676,\quad \theta_ 6\geq 0,5252,\quad \theta_ 7\geq 0,2821.\)