Some constructions of restricted Kakeya sets

@misc{chen2026some,
  title = {{Some constructions of restricted Kakeya sets}},
  author = {Cheng Chen and Jianbin Li and Longhui Li and Bochen Liu and Ruichen Sun and Haoran Wang and Saijie Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06556},
  eprint = {2609.06556},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06556},
}

TY  - UNPB
TI  - Some constructions of restricted Kakeya sets
AU  - Cheng Chen
AU  - Jianbin Li
AU  - Longhui Li
AU  - Bochen Liu
AU  - Ruichen Sun
AU  - Haoran Wang
AU  - Saijie Zhou
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06556
UR  - https://arxiv.org/abs/2609.06556
AB  - In this paper, we consider Kakeya sets with the additional restriction that centers of the unit line segments belong to a given set. In particular, for every uncountable Borel set A⊂ℝ^d, d≥ 2, we construct a compact subset of ℝ^d of Lebesgue measure zero that contains, in every direction, a unit line segment whose center lies in A. Notice that every such Kakeya set (not even necessarily compact) must have positive Lebesgue measure if A is countable. So our result shows that the countability is in fact the only obstruction.
ER  -