Box-counting by H\"older's traveling salesman

We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a H"older curve. This implies in particular that if the upper box-counting dimension of a set in a quasiconvex metric space is less or equal to $d\ge 1$, then for any $\alpha < \frac{1}{d}$thesetcanbecoveredbyan$\alpha$-H"older curve. On the other hand, for each $1\le d<2$ we give an example of a compact set $K$, in the plane, just failing the above Dini-type condition, with lower box-counting dimension equal to zero and upper box-counting dimension equal to $d$ that can not be covered by a countable collection of $\frac{1}{d}$-H"older curves.