@misc{ThreeDimensional,
  author = {{OpenAI}},
  title = {{The Kakeya maximal conjecture in three dimensions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf}{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026}},
  year = {2026},
  note = {Lemma~2.3}
}

@article{WangZahlSticky,
  author = {Wang, Hong and Zahl, Joshua},
  title = {Sticky {Kakeya} sets and the sticky {Kakeya} conjecture},
  journal = {Journal of the American Mathematical Society},
  volume = {39},
  number = {2},
  pages = {515--585},
  year = {2026},
  doi = {10.1090/jams/1067},
  eprint = {2210.09581},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Theorem numbering refers to arXiv:2210.09581v2, 7 October 2025},
  url = {https://arxiv.org/abs/2210.09581v2}
}

@article{CarberyValdimarsson,
  author = {Carbery, Anthony and Valdimarsson, Stef{\'a}n Ingi},
  title = {The endpoint multilinear {Kakeya} theorem via the {Borsuk--Ulam} theorem},
  journal = {Journal of Functional Analysis},
  volume = {264},
  number = {7},
  pages = {1643--1663},
  year = {2013},
  doi = {10.1016/j.jfa.2013.01.012},
  eprint = {1205.6371},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Theorem 1 on page 2 of arXiv:1205.6371v1, 29 May 2012},
  url = {https://arxiv.org/abs/1205.6371v1}
}

@misc{OrponenDeSaxceShmerkin,
  author = {Orponen, Tuomas and de Saxc{\'e}, Nicolas and Shmerkin, Pablo},
  title = {On the {Fourier} decay of multiplicative convolutions},
  year = {2024},
  howpublished = {arXiv:2309.03068v2},
  eprint = {2309.03068},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 2, 27 February 2024; Theorem 1.5},
  url = {https://arxiv.org/abs/2309.03068v2}
}

@book{TaoVu2006,
  author = {Tao, Terence and Vu, Van H.},
  title = {Additive Combinatorics},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {105},
  publisher = {Cambridge University Press},
  address = {Cambridge},
  year = {2006},
  doi = {10.1017/CBO9780511755149},
  note = {Theorem 2.29 and Exercise 6.4.10; see the author's \href{https://www.math.ucla.edu/~tao/preprints/books.html}{errata} for the corrected fifth-power loss}
}

@article{KatzZahlPlanebrush,
  author = {Katz, Nets Hawk and Zahl, Joshua},
  title = {A {Kakeya} maximal function estimate in four dimensions using planebrushes},
  journal = {Revista Matem{\'a}tica Iberoamericana},
  volume = {37},
  number = {1},
  pages = {317--359},
  year = {2021},
  doi = {10.4171/RMI/1219},
  eprint = {1902.00989},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 3, 7 October 2025; Theorem 2},
  url = {https://arxiv.org/abs/1902.00989v3}
}

@article{RaiChoudhuriSticky,
  author = {Rai Choudhuri, Mukul},
  title = {An improved bound on the {Hausdorff} dimension of sticky {Kakeya} sets in {$\mathbb{R}^4$}},
  journal = {Revista Matem{\'a}tica Iberoamericana},
  volume = {42},
  number = {4},
  pages = {1227--1256},
  year = {2026},
  doi = {10.4171/RMI/1621},
  url = {https://doi.org/10.4171/RMI/1621}
}

@misc{WangZakharovSticky,
  author = {Wang, Hong and Zakharov, Dmitrii},
  title = {{Kakeya} and {Furstenberg} problems for sticky sets of tubes},
  year = {2026},
  howpublished = {arXiv:2609.22035v1},
  eprint = {2609.22035},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 1, 18 September 2026; Theorem 1.2},
  url = {https://arxiv.org/abs/2609.22035v1}
}

@book{BasuPollackRoy,
  author = {Basu, Saugata and Pollack, Richard and Roy, Marie-Fran{\c{c}}oise},
  title = {Algorithms in Real Algebraic Geometry},
  edition = {2nd},
  series = {Algorithms and Computation in Mathematics},
  volume = {10},
  publisher = {Springer},
  address = {Berlin, Heidelberg},
  year = {2006},
  doi = {10.1007/3-540-33099-2},
  isbn = {978-3-540-33098-1},
  url = {https://doi.org/10.1007/3-540-33099-2}
}

@misc{WangZahlThreeDimensional,
  author = {Wang, Hong and Zahl, Joshua},
  title = {Volume estimates for unions of convex sets, and the {Kakeya} set conjecture in three dimensions},
  year = {2025},
  howpublished = {arXiv:2502.17655v1},
  eprint = {2502.17655},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 1, 24 February 2025},
  url = {https://arxiv.org/abs/2502.17655v1}
}

@misc{GuthWangZahlStreamlined,
  author = {Guth, Larry and Wang, Hong and Zahl, Joshua},
  title = {A streamlined proof of the {Kakeya} set conjecture in {$\mathbb{R}^3$}},
  year = {2026},
  howpublished = {arXiv:2601.14411v1},
  eprint = {2601.14411},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 1, 20 January 2026},
  url = {https://arxiv.org/abs/2601.14411v1}
}

@article{Besicovitch1928,
  author = {Besicovitch, A. S.},
  title = {On {Kakeya}'s problem and a similar one},
  journal = {Mathematische Zeitschrift},
  volume = {27},
  pages = {312--320},
  year = {1928},
  doi = {10.1007/BF01171101},
  url = {https://link.springer.com/article/10.1007/BF01171101}
}

@article{Davies1971,
  author = {Davies, Roy O.},
  title = {Some remarks on the {Kakeya} problem},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume = {69},
  number = {3},
  pages = {417--421},
  year = {1971},
  doi = {10.1017/S0305004100046867},
  url = {https://doi.org/10.1017/S0305004100046867}
}

@article{Wolff1995,
  author = {Wolff, Thomas H.},
  title = {An improved bound for {Kakeya} type maximal functions},
  journal = {Revista Matem{\'a}tica Iberoamericana},
  volume = {11},
  number = {3},
  pages = {651--674},
  year = {1995},
  doi = {10.4171/RMI/188},
  url = {https://ems.press/journals/rmi/articles/5248}
}

@misc{GaoLiuXi2025,
  author = {Gao, Chuanwei and Liu, Diankun and Xi, Yakun},
  title = {Curved {Kakeya} sets and {Nikodym} problems on manifolds},
  year = {2025},
  howpublished = {arXiv:2503.11574v1},
  eprint = {2503.11574},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  doi = {10.48550/arXiv.2503.11574},
  note = {Version 1, 14 March 2025; Definition 1.3, Conjecture 1.4, and Theorem 1.5; Definitions 1.7--1.11, Theorem 1.12, Corollary 4.2, and Section 4.2},
  url = {https://arxiv.org/abs/2503.11574v1}
}

@article{GuthZahlPolynomialWolff,
  author = {Guth, Larry and Zahl, Joshua},
  title = {Polynomial {Wolff} axioms and {Kakeya}-type estimates in {$\mathbb{R}^4$}},
  journal = {Proceedings of the London Mathematical Society},
  volume = {117},
  number = {1},
  pages = {192--220},
  year = {2018},
  doi = {10.1112/plms.12138},
  eprint = {1701.07045},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Corrected version: arXiv:1701.07045v3, 19 April 2019},
  url = {https://arxiv.org/abs/1701.07045v3}
}

@article{ZahlSeveri,
  author = {Zahl, Joshua},
  title = {A discretized {Severi}-type theorem with applications to harmonic analysis},
  journal = {Geometric and Functional Analysis},
  volume = {28},
  pages = {1131--1181},
  year = {2018},
  doi = {10.1007/s00039-018-0455-x},
  eprint = {1801.05106},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  url = {https://arxiv.org/abs/1801.05106v3}
}

@article{KatzRogersPolynomialWolff,
  author = {Katz, Nets Hawk and Rogers, Keith M.},
  title = {On the polynomial {Wolff} axioms},
  journal = {Geometric and Functional Analysis},
  volume = {28},
  number = {6},
  pages = {1706--1716},
  year = {2018},
  doi = {10.1007/s00039-018-0466-7},
  eprint = {1802.09094},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  url = {https://arxiv.org/abs/1802.09094v3}
}

@article{RoytwarfYomdin1997,
  author = {Roytwarf, N. and Yomdin, Y.},
  title = {{Bernstein} classes},
  journal = {Annales de l'Institut Fourier},
  volume = {47},
  number = {3},
  pages = {825--858},
  year = {1997},
  doi = {10.5802/aif.1582},
  url = {https://doi.org/10.5802/aif.1582}
}

@book{CoverThomas2006,
  author = {Cover, Thomas M. and Thomas, Joy A.},
  title = {Elements of Information Theory},
  edition = {2nd},
  publisher = {Wiley},
  year = {2006},
  doi = {10.1002/047174882X},
  url = {https://doi.org/10.1002/047174882X}
}

@article{SudakovSzemerediVu2005,
  author = {Sudakov, Benny and Szemer{\'e}di, Endre and Vu, Van H.},
  title = {On a question of {Erd\H{o}s} and {Moser}},
  journal = {Duke Mathematical Journal},
  volume = {129},
  number = {1},
  pages = {129--155},
  year = {2005},
  doi = {10.1215/S0012-7094-04-12915-X},
  url = {https://doi.org/10.1215/S0012-7094-04-12915-X}
}

@article{BennettCarberyTao2006,
  author = {Bennett, Jonathan and Carbery, Anthony and Tao, Terence},
  title = {On the multilinear restriction and {Kakeya} conjectures},
  journal = {Acta Mathematica},
  volume = {196},
  number = {2},
  pages = {261--302},
  year = {2006},
  doi = {10.1007/s11511-006-0006-4},
  url = {https://doi.org/10.1007/s11511-006-0006-4}
}

@article{Guth2010,
  author = {Guth, Larry},
  title = {The endpoint case of the {Bennett--Carbery--Tao} multilinear {Kakeya} conjecture},
  journal = {Acta Mathematica},
  volume = {205},
  number = {2},
  pages = {263--286},
  year = {2010},
  doi = {10.1007/s11511-010-0055-6},
  url = {https://doi.org/10.1007/s11511-010-0055-6}
}

@article{BourgainGuth2011,
  author = {Bourgain, Jean and Guth, Larry},
  title = {Bounds on oscillatory integral operators based on multilinear estimates},
  journal = {Geometric and Functional Analysis},
  volume = {21},
  number = {6},
  pages = {1239--1295},
  year = {2011},
  doi = {10.1007/s00039-011-0140-9},
  url = {https://doi.org/10.1007/s00039-011-0140-9}
}

@article{GuoWangZhangDichotomy,
  author = {Guo, Shaoming and Wang, Hong and Zhang, Ruixiang},
  title = {A dichotomy for {H{\"o}rmander}-type oscillatory integral operators},
  journal = {Inventiones mathematicae},
  volume = {238},
  number = {2},
  pages = {503--584},
  year = {2024},
  doi = {10.1007/s00222-024-01288-8},
  eprint = {2210.05851},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {The invariant formulation is Theorem 2.1 in arXiv:2210.05851v2, 3 February 2023},
  url = {https://arxiv.org/abs/2210.05851v2}
}

@article{NadjimzadahBourgain,
  author = {Nadjimzadah, Arian},
  title = {{Bourgain}'s Condition, Sticky {Kakeya}, and New Examples},
  journal = {Geometric and Functional Analysis},
  volume = {36},
  pages = {1131--1173},
  year = {2026},
  doi = {10.1007/s00039-026-00750-4},
  eprint = {2511.10918},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Example 1.24 and Proposition 1.25; arXiv:2511.10918v4, 18 August 2026},
  url = {https://arxiv.org/abs/2511.10918v4}
}

@article{Bourgain1999,
  author = {Bourgain, Jean},
  title = {On the dimension of {Kakeya} sets and related maximal inequalities},
  journal = {Geometric and Functional Analysis},
  volume = {9},
  number = {2},
  pages = {256--282},
  year = {1999},
  doi = {10.1007/s000390050087},
  url = {https://link.springer.com/article/10.1007/s000390050087}
}

@article{KatzLabaTao2000,
  author = {Katz, Nets Hawk and {\L}aba, Izabella and Tao, Terence},
  title = {An improved bound on the {Minkowski} dimension of {Besicovitch} sets in {$\mathbb{R}^3$}},
  journal = {Annals of Mathematics},
  volume = {152},
  number = {2},
  pages = {383--446},
  year = {2000},
  doi = {10.2307/2661389},
  eprint = {math/9903166},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  url = {https://arxiv.org/abs/math/9903166v2}
}

@article{KatzTaoNewBounds,
  author = {Katz, Nets Hawk and Tao, Terence},
  title = {New bounds for {Kakeya} problems},
  journal = {Journal d'Analyse Math{\'e}matique},
  volume = {87},
  pages = {231--263},
  year = {2002},
  doi = {10.1007/BF02868476},
  eprint = {math/0102135},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  url = {https://arxiv.org/abs/math/0102135}
}

@misc{KeletiMatheEquivalences,
  author        = {Keleti, Tam{\'a}s and M{\'a}th{\'e}, Andr{\'a}s},
  title         = {Equivalences between different forms of the {Kakeya}
                   conjecture and duality of {Hausdorff} and packing
                   dimensions for additive complements},
  year          = {2023},
  eprint        = {2203.15731},
  archivePrefix = {arXiv},
  primaryClass  = {math.MG},
  note          = {Version 3, 17 July 2023},
  url           = {https://arxiv.org/abs/2203.15731v3}
}

@misc{NadjimzadahLifting,
  author        = {Nadjimzadah, Arian},
  title         = {Lifting curved {Kakeya} sets to linear {Kakeya} sets},
  year          = {2026},
  eprint        = {2609.06299},
  archivePrefix = {arXiv},
  primaryClass  = {math.CA},
  note          = {Version 1, 5 September 2026},
  url           = {https://arxiv.org/abs/2609.06299v1}
}

@article{LabaTaoMediumDimension,
  author = {{\L}aba, Izabella and Tao, Terence},
  title = {An improved bound for the {Minkowski} dimension of {Besicovitch} sets in medium dimension},
  journal = {Geometric and Functional Analysis},
  volume = {11},
  pages = {773--806},
  year = {2001},
  doi = {10.1007/PL00001685},
  eprint = {math/0004015},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Theorem 1.3 in arXiv:math/0004015v1},
  url = {https://arxiv.org/abs/math/0004015v1}
}

@misc{BorgesEtAlRestrictionKakeya,
  author = {Borges, Tainara and Chan, Tiklung and Chen, Mingfeng and Liu, Diankun and Xi, Yakun and Zhan, Yufei},
  title = {Restriction and {Kakeya} maximal estimates in {$\mathbb{R}^4$}},
  year = {2025},
  howpublished = {arXiv:2511.22824v1},
  eprint = {2511.22824},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 1, 28 November 2025; Theorem 1.2},
  url = {https://arxiv.org/abs/2511.22824v1}
}

@misc{ChristQuasiextremals,
  author = {Christ, Michael},
  title = {Quasiextremals for a {Radon}-like transform},
  year = {2011},
  howpublished = {arXiv:1106.0722v1},
  eprint = {1106.0722},
  archivePrefix = {arXiv},
  primaryClass = {math.CA},
  note = {Version 1, 3 June 2011},
  url = {https://arxiv.org/abs/1106.0722v1}
}