Typical Kakeya-type sets and their typical code sets
Abstract
We study properties of compact Besicovitch sets and other Kakeya-type sets which are typical in the sense that the exceptions form a meager set in the Baire category sense in a suitable metric space. The main motivation comes from the known fact that typical compact Besicovitch sets in ℝn have minimal Hausdorff dimension among all Besicovitch sets of ℝn. Thus, in order to estimate the smallest possible Hausdorff dimension of an arbitrary Besicovitch set, it is enough to estimate the Hausdorff dimension of a typical (compact) Besicovitch set, or analogously of a Besicovitch set satisfying our list of typical properties. Among other results, we prove that a typical Besicovitch set K in ℝn has the following properties: it is not sticky, any line intersects K in a set of ($1$-dimensional) measure zero or in a unit line segment and a set of measure zero, it has no more than n different unit line segments in any direction, and for each 1≤ j≤ n the Hausdorff dimension of the set of those directions in which K has exactly j unit line segments is n−j. For compact metric spaces X and Y, let 𝒞(X,Y) denote the set of compact subsets of X× Y whose projection onto the first coordinate is X, equipped with the Hausdorff metric. Kakeya-type compact sets can be coded by elements of such spaces, allowing typical properties to be studied through comeager subsets of 𝒞(X,Y). Among other results, we prove that, under suitable conditions on X and Y, a typical A∈ 𝒞(X,Y) is totally disconnected and purely unrectifiable, its projection to the second coordinate has Hausdorff dimension zero, and the dimension identities dimH(A)=dimH(X) and dimP(A)=dimP(X)+dim Y hold.
Related
BibTeX
@misc{bell2026typical,
title = {{Typical Kakeya-type sets and their typical code sets}},
author = {Josef Bell and Ezekiel Fleissner-Kates and Tamás Keleti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37419},
eprint = {2610.37419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37419},
}
RIS
TY - UNPB
TI - Typical Kakeya-type sets and their typical code sets
AU - Josef Bell
AU - Ezekiel Fleissner-Kates
AU - Tamás Keleti
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37419
UR - https://arxiv.org/abs/2610.37419
AB - We study properties of compact Besicovitch sets and other Kakeya-type sets which are typical in the sense that the exceptions form a meager set in the Baire category sense in a suitable metric space. The main motivation comes from the known fact that typical compact Besicovitch sets in ℝ^n have minimal Hausdorff dimension among all Besicovitch sets of ℝ^n. Thus, in order to estimate the smallest possible Hausdorff dimension of an arbitrary Besicovitch set, it is enough to estimate the Hausdorff dimension of a typical (compact) Besicovitch set, or analogously of a Besicovitch set satisfying our list of typical properties. Among other results, we prove that a typical Besicovitch set K in ℝ^n has the following properties: it is not sticky, any line intersects K in a set of ($1$-dimensional) measure zero or in a unit line segment and a set of measure zero, it has no more than n different unit line segments in any direction, and for each 1≤ j≤ n the Hausdorff dimension of the set of those directions in which K has exactly j unit line segments is n−j. For compact metric spaces X and Y, let 𝒞(X,Y) denote the set of compact subsets of X× Y whose projection onto the first coordinate is X, equipped with the Hausdorff metric. Kakeya-type compact sets can be coded by elements of such spaces, allowing typical properties to be studied through comeager subsets of 𝒞(X,Y). Among other results, we prove that, under suitable conditions on X and Y, a typical A∈ 𝒞(X,Y) is totally disconnected and purely unrectifiable, its projection to the second coordinate has Hausdorff dimension zero, and the dimension identities dim_H(A)=dim_H(X) and dim_P(A)=dim_P(X)+dim Y hold.
ER -