H\"older coverings of sets of small dimension

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.