The paper is devoted to the relationship between the property of a set in a metric space to be covered by a Hölder curve and the box-counting dimension. More precisely, let \(Y\) be a subset of a complete, quasiconvex metric space \(X\), and \({N(Y,\varepsilon)}\) denotes the minimal number of balls of radius \({\varepsilon>0}\) needed to cover \(Y.\) The first main result states that if for some \({d\geq1}\) and \({\varepsilon_0>0}\) there is the Dini condition \[ \sum_{k\geq0}N(Y,\varepsilon_02^{-k})2^{-kd}<\infty, \] then \(Y\) can be covered by a \(1/d\)-Hölder curve. In the Euclidean space \({\mathbb{R}^n}\) a much stronger result is due to M. Badger et al. [Adv. Math. 349, 564–647 (2019; Zbl 1411.28001)].
The second main result is the construction of a compact set \(K\), for each \({d\in[1,2)}\), in the plane with the following properties: 1) the lower box-counting dimension of \(K\) is zero and the upper box-counting dimension of \(K\) equals \(d\); 2) the Dini-type condition for \(K\) does not hold; 3) \(K\) can not be covered even not by a countable collection of \(1/d\)-Hölder curves.
The construction of \(K\) uses a modification of the construction of the standard four corner 1/4 Cantor set. The proof of some properties relies on a result of P. W. Jones [Invent. Math. 102, No. 1, 1–15 (1990; Zbl 0731.30018)].