Hausdorff measures of subgroups of \(\mathbb{R}/\mathbb{Z}\) and \(\mathbb{R}\)
Let \(u=(u_n)_{n\in{\mathbb N}}\) be a sequence in \(]0, +\infty[\). The authors study the Hausdorff measure of the group \[ \tau_u =\{x\in{\mathbb R}: u_{n}x\to 0 \quad \text{modulo} \quad 1\}. \] Let \({\mathcal H}^f\) stand for the Hausdorff measure generated by an increasing function \(f\). The main result of the paper is the construction of the sequence \(u\) such that \({\mathcal H}^{f}(\tau_u)=+\infty\) and \({\mathcal H}^{g}(\tau_u)=0\) for a given pair of functions \(f\), \(g\) such that \(f(t)/g(t) \to +\infty\), \(f(t)/t \to +\infty\) and \(g(t)/t \to +\infty\) for \(t\to +0\).
scientific article; zbMATH DE number 6378221