Box-counting by Hölder’s traveling salesman |
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Authors: | Zolt n M. Balogh,Roger Z st |
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Affiliation: | Zoltán M. Balogh,Roger Züst |
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Abstract: | We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a Hölder curve. This implies in particular that if the upper box-counting dimension is less than $$d ge 1$$, then it can be covered by an $$frac{1}{d}$$-Hölder curve. On the other hand, for each $$1le d <2$$ we give an example of a compact set in the plane with lower box-counting dimension equal to zero and upper box-counting dimension equal to d, just failing the above Dini-type condition, that can not be covered by a countable collection of $$frac{1}{d}$$-Hölder curves. |
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