Editing Kakeya set (section)

==Kakeya conjecture==

===Statement===
The same question of how small these Besicovitch sets could be was then posed in higher dimensions, giving rise to a number of conjectures known collectively as the ''Kakeya conjectures'', and have helped initiate the field of mathematics known as [[geometric measure theory]].  In particular, if there exist Besicovitch sets of measure zero, could they also have s-dimensional [[Hausdorff measure]] zero for some dimensions less than the dimension of the space in which they lie? This question gives rise to the following conjecture:

:'''Kakeya set conjecture''': a set in Euclidean space that contains a unit line segment in every direction must have a Hausdorff dimension equal to the dimension of the space. 

This is known to be true for ''n'' = 1, 2 but only partial results are known in higher dimensions.

In February 2025, a claimed proof for the case ''n'' = 3 was posted on [[arXiv]] by [[Hong Wang (mathematician)|Hong Wang]] and Joshua Zahl.<ref name=":1">{{cite arXiv |eprint=2502.17655 |class=math.CA |author1=Hong Wang |author2=Joshua Zahl |title=Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions |date=2025-02-24}}</ref> The Kakeya conjecture in three dimensions is described as "one of the most sought-after open problems in geometric measure theory", and the claimed proof is considered to be a breakthrough.<ref>{{cite web |title=Chinese maths star Wang Hong solves 'infamous' geometry problem |url=https://www.scmp.com/news/china/science/article/3300958/chinese-maths-star-wang-hong-solves-infamous-geometry-problem?module=perpetual_scroll_0&pgtype=article |archive-url=https://archive.today/20250304114650/https://www.scmp.com/news/china/science/article/3300958/chinese-maths-star-wang-hong-solves-infamous-geometry-problem |archive-date=4 March 2025 |date=4 March 2025|publisher=[[South China Morning Post]]}}</ref><ref>{{cite web |title=Century-Old Math Enigma Finally Solved: How Chinese Student Cracked An 'Impossible' Geometry Mystery |url=https://english.jagran.com/world/chinese-student-wang-hong-cracked-an-impossible-geometry-puzzle-that-stumped-mathematicians-for-over-a-century-10221886 |website=[[Dainik Jagran]] |archive-url=https://archive.today/20250304154738/https://english.jagran.com/world/chinese-student-wang-hong-cracked-an-impossible-geometry-puzzle-that-stumped-mathematicians-for-over-a-century-10221886 |archive-date=4 March 2025 |date=4 March 2025}}</ref><ref>{{Cite web |last=Howlett |first=Joseph |date=2025-03-14 |title='Once in a Century' Proof Settles Math's Kakeya Conjecture |url=https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/ |access-date=2025-03-21 |website=Quanta Magazine |language=en}}</ref>

===Kakeya maximal function===
A modern way of approaching this problem is to consider a particular type of [[maximal function]], which we construct as follows: Denote '''S'''<sup>''n''−1</sup> ⊂ '''R'''<sup>''n''</sup> to be the unit sphere in ''n''-dimensional space. Define <math>T_{e}^{\delta}(a)</math> to be the cylinder of length 1, radius δ > 0, centered at the point ''a'' ∈ '''R'''<sup>''n''</sup>, and whose long side is parallel to the direction of the unit vector ''e'' ∈ '''S'''<sup>''n''−1</sup>. Then for a [[locally integrable]] function ''f'', we define the '''Kakeya maximal function''' of ''f'' to be

:<math> f_{*}^{\delta}(e)=\sup_{a\in\mathbf{R}^{n}}\frac{1}{m(T_{e}^{\delta}(a))}\int_{T_{e}^{\delta}(a)}|f(y)|dm(y),</math>

where ''m'' denotes the ''n''-dimensional [[Lebesgue measure]]. Notice that <math>f_{*}^{\delta}</math> is defined for vectors ''e'' in the sphere '''S'''<sup>''n''−1</sup>.

Then there is a conjecture for these functions that, if true, will imply the Kakeya set conjecture for higher dimensions:

:'''Kakeya maximal function conjecture''': For all ε > 0, there exists a constant ''C<sub>ε</sub>'' > 0 such that for any function ''f'' and all δ > 0, (see [[lp space]] for notation)
::<math> \left \|f_{*}^{\delta} \right \|_{L^n(\mathbf{S}^{n-1})} \leqslant C_{\epsilon} \delta^{-\epsilon}\|f\|_{L^n(\mathbf{R}^{n})}. </math>

=== Results===
Some results toward proving the Kakeya conjecture are the following:
* The Kakeya conjecture is true for ''n'' = 1 (trivially) and ''n'' = 2 (Davies<ref>{{cite journal| last = Davies | first = Roy  | title =Some remarks on the Kakeya problem | journal =[[Mathematical Proceedings of the Cambridge Philosophical Society]] | volume = 69 | pages = 417&ndash;421 | year = 1971  | doi = 10.1017/S0305004100046867 | issue = 3| bibcode = 1971PCPS...69..417D}}</ref>).
* In any ''n''-dimensional space, Wolff<ref>{{cite journal  | last = Wolff | first = Thomas | author-link = Thomas Wolff| title = An improved bound for Kakeya type maximal functions  | journal = Rev. Mat. Iberoamericana | volume = 11 | pages = 651&ndash;674 | year = 1995 | issue = 3 | doi=10.4171/rmi/188| doi-access = free }}</ref> showed that the dimension of a Kakeya set must be at least (''n''+2)/2.
* In 2002, [[Nets Hawk Katz|Katz]] and [[Terence Tao|Tao]]<ref>{{cite journal | last1=Katz | first1=Nets Hawk| authorlink1=Nets Hawk Katz|last2=Tao | first2=Terence | authorlink2=Terence Tao | title = New bounds for Kakeya problems  | journal = [[Journal d'Analyse Mathématique]] | volume = 87 | pages = 231&ndash;263 | year = 2002 | doi=10.1007/BF02868476|doi-access=free| arxiv=math/0102135| s2cid=119644987}}</ref> improved Wolff's bound to <math>(2-\sqrt{2})(n-4)+3</math>, which is better for ''n'' > 4.
* In 2000, Katz,  [[Izabella Łaba|Łaba]], and Tao<ref>{{cite journal|last1=Katz|first1=Nets Hawk|last2=Łaba|first2=Izabella|last3=Tao|first3=Terence|title=An Improved Bound on the Minkowski Dimension of Besicovitch Sets in <math>\mathbb{R}^3</math>|journal=The Annals of Mathematics|date=September 2000|volume=152|issue=2|pages=383&ndash; 446 |doi=10.2307/2661389|jstor=2661389|arxiv=math/0004015|s2cid=17007027}}</ref> proved that the [[Minkowski–Bouligand dimension|Minkowski dimension]] of Kakeya sets in 3 dimensions is strictly greater than 5/2.
* In 2000, [[Jean Bourgain]] connected the Kakeya problem to [[arithmetic combinatorics]]<ref>J. Bourgain, Harmonic analysis and combinatorics: How much may they contribute to each other?, Mathematics: Frontiers and Perspectives, IMU/Amer. Math. Soc., 2000, pp. 13–32.</ref><ref>{{cite journal | last = Tao | first = Terence | author-link = Terence Tao  | title = From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis and PDE| url=https://www.ams.org/notices/200103/fea-tao.pdf | journal = Notices of the AMS | volume = 48 | issue = 3 | pages = 297&ndash;303 |date=March 2001}}</ref> which involves [[harmonic analysis]] and [[additive number theory]].
* In 2017, Katz and Zahl<ref>{{cite journal|last1=Katz|first1=Nets Hawk|last2=Zahl|first2=Joshua|title=An improved bound on the Hausdorff dimension of Besicovitch sets in <math>\mathbb{R}^3</math>|journal=Journal of the American Mathematical Society|date=2019|volume=32|issue=1|pages= 195&ndash; 259 |doi=10.1090/jams/907 |arxiv=1704.07210|s2cid=119322412}}</ref> improved the lower bound on the [[Hausdorff dimension]] of Besicovitch sets in 3 dimensions to <math>5/2+\epsilon</math> for an absolute constant <math>\epsilon>0</math>.
* In 2025, Wang and Zahl<ref name=":1" /> posted on [[arXiv]] a potential proof of the Kakeya conjecture in the case n = 3.