On the exact Hausdorff dimension of the set of Liouville numbers. II (Q818643)

This paper extends a previous resultby the first author [cf. Part I, Manuscr. Math. 116, No.~2, 157--172 (2005; Zbl 1095.28012)] on the classification of dimension functions for which the associated \(h\)-Hausdorff measure of the set of Liouville numbers \(L\) is either null or infinite. The authors prove here that for any arbitrary dimension function \(h\), either \(L\) has zero \(h\)-Hausdorff measure or \(L\) cannot be decomposed in countably many sets of finite \(h\)-Hausdorff measure (in particular, \(L\) has infinite \(h\)-Hausdorff measure). The dichotomy depends on the potential growth of the function \(\Gamma(r):=\inf_{0<s\leq r}rh(s)/s\) as \(r\) goes to 0.