Box-counting by Hölder’s traveling salesman

Abstract

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 of a set in a quasiconvex metric space is less or equal to d ≥ 1, then for any α < 1 d the set can be covered by an α-Hölder curve. On the other hand, for each 1 ≤ 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 1 d -Hölder curves.