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

## TL;DR

This paper establishes conditions under which sets in quasiconvex metric spaces can be covered by H"older curves, linking box-counting dimensions with H"older regularity, and provides counterexamples for certain dimensions.

## Contribution

It introduces a Dini-type condition for covering sets with H"older curves and demonstrates its sharpness through counterexamples in the plane.

## Key findings

- Sets with upper box-counting dimension ≤ d can be covered by α-H"older curves for α<1/d.
- Counterexamples show limitations of covering by H"older curves when conditions are not met.
- The paper connects fractal dimensions with geometric covering properties in metric spaces.

## Abstract

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 $d \geq 1$, then for any $\alpha < \frac{1}{d}$ the set can be covered by an $\alpha$-H\"older curve. On the other hand, for each $1\leq d <2$ 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.