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PreprintarXiv licencearXiv

Dimension reduction by bi-Hölder maps

  • Leonid V. Kovalev
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Abstract

The Hölder dimension of a metric space is the infimum of its Hausdorff dimension over all bi-Hölder changes of the metric. Colvin asked whether it can take noninteger values. We show that every metric space of finite upper box dimension has integer Hölder dimension. For a bounded subset of a Euclidean space, the dimension reduction is carried out by a global homeomorphism of the ambient space, built from randomly translated singular functions, which brings the Hausdorff dimension of the set arbitrarily close to its integer part.

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Cite

BibTeXRIS
BibTeX
@misc{kovalev2026dimension,
  title = {{Dimension reduction by bi-Hölder maps}},
  author = {Leonid V. Kovalev},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.11482},
  eprint = {2611.11482},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.11482},
}
RIS
TY  - UNPB
TI  - Dimension reduction by bi-Hölder maps
AU  - Leonid V. Kovalev
PY  - 2026
DA  - 2026/11/08
PB  - arXiv
DO  - 10.48550/arxiv.2611.11482
UR  - https://arxiv.org/abs/2611.11482
AB  - The Hölder dimension of a metric space is the infimum of its Hausdorff dimension over all bi-Hölder changes of the metric. Colvin asked whether it can take noninteger values. We show that every metric space of finite upper box dimension has integer Hölder dimension. For a bounded subset of a Euclidean space, the dimension reduction is carried out by a global homeomorphism of the ambient space, built from randomly translated singular functions, which brings the Hausdorff dimension of the set arbitrarily close to its integer part.
ER  -