Vectors of a given Diophantine type. II

We prove the existence of a family of vectors with continuum many elements $\mathbf v\in\mathbb{R}^s$ admitting infinitely many simultaneous $(\varphi(p)/p^{1/s})(1+B\cdot\varphi^{1+1/s}(p))$-approximations and admitting no simultaneous $(\varphi(p)/p^{1/s})(1-B\cdot\varphi^{1+1/s}(p))$-approximation.
We prove that for $0<t\le T$ the closed interval $[t,t(1+16B\cdot t^{1+1/s})]$ contains an element of the $s$-dimensional Lagrange spectrum. Here $A$, $B$ and $T$ stand for some positive constants depending on the dimension $s$ only and $\varphi$ is a positive nonincreasing function of positive integer argument such that $\varphi(1)\le A$.
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