A remark on the set of exactly approximable vectors in the simultaneous case (Q6566370)

Let \(n\) be a positive integer and \(\psi :\mathbb N\to (0,1]\) be such that \(\lim_{q\to\infty} q^2\psi(q)=0\). Assume that upper and lower orders at infinity of the function \(1/\psi\) coincide and are equal to \(\lambda \geq 2\). Let \(W(n,\psi)\) be the set of all \((\alpha_1,\dots ,\alpha_n)\in [0,1]^n\) such that \(\max_{i=1,\dots ,n}|\alpha_i-\frac {p_i}q |<\psi(q)\) for infinitely many \((q,p_1,\dots ,p_n)\in \mathbb N\times \mathbb Z^n\). Set \(E(n,\psi)=W(n,\psi)\setminus \cup_{0<c<1} W(n,c\psi)\).\N\NThen the author proves that for \(n\geq 3\) we have \(\dim \ E(n,\psi)=\dim \ W(n,\psi)=\frac {n+1}{\lambda}\), where \(\dim\) denotes the Hausdorff dimension.