On absolutely friendly measures on \(\mathbb{Q}_S^d\) (Q6581859)

Let \(d\) be a positive integer, \(\mu\) be a \(\alpha\)-absolutely friendly measure on a compact set and \(S\) be a finite set of valuations of \(\mathbb Q\) with respect to \(\mu\). Let \(\psi :\mathbb N\to\mathbb R^+\) be a monotonically decreasing function. Set \(\mathbb Q_S^d=\prod_{j\in S} \mathbb Q_j^d\). Let \(l\) be the number of elements of \(S\). Let \(\Vert \cdot \Vert_S\) be the max norm. For \(\infty\notin S\) let \(W^S(\psi)\) be all numbers from \(x\in \mathbb Q_S^d\) such that for infinitely many \((q,q_0)\in\mathbb Z^{d+1}\) we have \(\Vert x +\frac q{q_0}\Vert_S^l\leq \psi(\Vert (q,q_0)\Vert_\infty)\), and for \(\infty\in S\) let \(W^S(\psi)\) be all numbers from \(x\in \mathbb Q_S^d\) such that for infinitely many \((q,q_0)\in\mathbb Z^{d+1}\) we have \(\Vert x +\frac q{q_0}\Vert_S^l\leq \psi(\vert (q,q_0)\vert_\infty)\). Then the authors prove that \(\mu(W^S(\psi)\cap supp \ \mu)=0\) if \(\sum_{n\in\mathbb N}(2^{n\frac{d+1}d}\psi(2^n))^{\alpha}<\infty\).