Small Furstenberg sets
Ursula Molter, Ezequiel Rela

TL;DR
This paper constructs Furstenberg sets with refined Hausdorff measure bounds, improving previous results and establishing sharp dimension estimates for zero-dimensional Furstenberg sets.
Contribution
It introduces new constructions of Furstenberg sets with precise Hausdorff measure bounds, advancing understanding of their dimensional properties.
Findings
Existence of Furstenberg sets with zero Hausdorff measure for specific gauge functions.
Sharp Hausdorff dimension estimate of 1/2 for a class of zero-dimensional Furstenberg sets.
Improved bounds on Hausdorff measure for Furstenberg sets with positive alpha.
Abstract
For in , a subset of is called Furstenberg set of type or -set if for each direction in the unit circle there is a line segment in the direction of such that the Hausdorff dimension of the set is greater or equal than . In this paper we show that if , there exists a set such that for , , which improves on the the previously known bound, that for . Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for , , we construct a set of Hausdorff dimension not…
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Taxonomy
TopicsMathematical Approximation and Integration · Digital Image Processing Techniques · Computational Geometry and Mesh Generation
