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 $d\ge 1$, then for any $\alpha <\frac{1}{d}$ the set can be covered by an $\alpha$-Hölder 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 $\frac{1}{d}$-Hölder curves.

11 pages