On simultaneous Diophantine approximations. Vectors of given Diophantine type

For any monotone function $\psi(y)=O(y^{-1/s})$, we prove the existence of a continual family of vectors $(\alpha_1,\dots,\alpha_s)\in\mathbb R^s$ admitting infinitely many simultaneous $\psi$-approximations, but no $c\psi$-approximations with some constant $c>0$.