# H\"older coverings of sets of small dimension

## TL;DR

This paper demonstrates that small-dimensional sets can be covered by H"older graphs in most directions, providing sharp bounds on the exceptional set and exploring implications for measure and coordinate systems.

## Contribution

It improves bounds on H"older coverings of small-dimensional sets and addresses measure and coordinate system questions related to these coverings.

## Key findings

- Sets of small box counting dimension can be covered by H"older graphs in most directions.
- Sharp bounds are established for the dimension of the exceptional set of directions.
- H"older graphs can have positive doubling measure, answering an open question.

## Abstract

We show that a set of small box counting dimension can be covered by a H\"older graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We observe that, as a consequence, H\"older graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on H\"older coverings in polar coordinates and, on the other hand, prove that a Homogenous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions.