Vectors of Given Diophantine Type

For a function $\varphi(y)=o(y^{-1/s})$, $y\to\infty$, we prove the existence of vectors $\overline\alpha\in\mathbb R^s$ admitting, for any $\varepsilon>0$, infinitely many simultaneous $\varphi(1+\varepsilon)$-approximations, but not admitting any simultaneous $\varphi$-approximations.