Lifting curved Kakeya sets to linear Kakeya sets

@misc{nadjimzadah2026lifting,
  title = {{Lifting curved Kakeya sets to linear Kakeya sets}},
  author = {Arian Nadjimzadah},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.06299},
  eprint = {2609.06299},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06299},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Lifting curved Kakeya sets to linear Kakeya sets
AU  - Arian Nadjimzadah
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.06299
UR  - https://arxiv.org/abs/2609.06299
AB  - We prove that many curved Kakeya problems, originally motivated by Hörmander's oscillatory integral problem, lift to the classical Kakeya problem in higher dimensions. As a consequence, if the classical Kakeya set conjecture were true in all dimensions, the families of curves whose curved Kakeya sets have full dimension are dense among Hörmander-type families. This is in contrast to the nowhere dense Bourgain's condition, which is a necessary condition for best-case curved Kakeya maximal function estimates. Furthermore the linear Kakeya set conjecture would imply a fairly complete understanding of Kakeya sets of quadratic curves, as first studied systematically by Wisewell. These results give a better understanding of a question posed by Guo--Guth--Nadjimzadah--Shen--Zhang, and they more generally show that the classical Kakeya conjecture in high dimensions rests on a broad range of curved Kakeya problems in lower dimensions.
ER  -