Hölder coverings of sets of small dimension

  • Eino Rossi

    University of Helsinki, Finland
  • Pablo Shmerkin

    Torcuato di Tella University, Buenos Aires, Argentina
Hölder coverings of sets of small dimension cover

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Abstract

We show that a set of small box counting dimension can be covered by a Hölder 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, Hölder graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on Hölder coverings in polar coordinates and, on the other hand, prove that a homogeneous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions.

Cite this article

Eino Rossi, Pablo Shmerkin, Hölder coverings of sets of small dimension. J. Fractal Geom. 6 (2019), no. 3, pp. 285–299

DOI 10.4171/JFG/78