Box-counting by Hölder’s traveling salesman

We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a Hölder curve. This implies in particular that if the upper box-counting dimension is less than d ≥ 1 , then it can be covered by an 1 d -Hölder curve. On the other hand, for each 1 ≤ d < 2 we give an example of a compact set in the plane with lower box-counting dimension equal to zero and upper box-counting dimension equal to d , just failing the above Dini-type condition, that can not be covered by a countable collection of 1 d -Hölder curves.