Kakeya set#Kakeya sets in vector spaces over finite fields

The Kakeya needle problem asks whether there is a minimum area of a region $D$ in the plane, in which a needle of unit length can be turned through 360°. This question was first posed, for convex regions, by {{harvs|txt|author-link=Sōichi Kakeya|first=Sōichi|last= Kakeya|year=1917}}. The minimum area for convex sets is achieved by an equilateral triangle of height 1 and area 1/{{radic|3}}, as Pál showed.{{cite journal| last = Pal | first = Julius | title = Ueber ein elementares variationsproblem | journal = Kongelige Danske Videnskabernes Selskab Math.-Fys. Medd. | volume = 2 | pages = 1–35 | year = 1920 }}

Kakeya seems to have suggested that the Kakeya set $D$ of minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false; there are smaller non-convex Kakeya sets.

Image:Perron tree.svg

Besicovitch was able to show that there is no lower bound > 0 for the area of such a region $D$, in which a needle of unit length can be turned around. That is, for every $\backslash varepsilon>0$, there is region of area $\backslash varepsilon$ within which the needle can move through a continuous motion that rotates it a full 360 degrees.{{cite journal | last=Besicovitch | first=Abram | author-link=Abram Samoilovitch Besicovitch | title=Sur deux questions d'integrabilite des fonctions | journal=J. Soc. Phys. Math. | pages=105–123 | volume=2 | year=1919}}
{{cite journal | last=Besicovitch | first=Abram | author-link=Abram Samoilovitch Besicovitch | title=On Kakeya's problem and a similar one | journal=Mathematische Zeitschrift | volume=27 | pages=312–320 | year=1928 | doi=10.1007/BF01171101| s2cid=121781065 }} This built on earlier work of his, on plane sets which contain a unit segment in each orientation. Such a set is now called a Besicovitch set. Besicovitch's work from 1919 showed such a set could have arbitrarily small measure, although the problem may have been considered by analysts before that.

One method of constructing a Besicovitch set (see figure for corresponding illustrations) is known as a "Perron tree", named after Oskar Perron who was able to simplify Besicovitch's original construction.{{cite journal | last=Perron | first=O. | title=Über einen Satz von Besicovitch | journal=Mathematische Zeitschrift | volume=28 | pages=383–386 | year=1928 | doi=10.1007/BF01181172| s2cid=120768630 }}
{{cite book | last=Falconer | first=K. J. | title=The Geometry of Fractal Sets | publisher=Cambridge University Press | year=1985 | pages=96–99}} The precise construction and numerical bounds are given in Besicovitch's popularization.

The first observation to make is that the needle can move in a straight line as far as it wants without sweeping any area. This is because the needle is a zero width line segment. The second trick of Pál, known as Pál joins,[http://www.mathematik.uni-muenchen.de/~lerdos/Stud/furtner.pdf The Kakeya Problem] {{Webarchive|url=https://web.archive.org/web/20150715012335/http://www.mathematik.uni-muenchen.de/~lerdos/Stud/furtner.pdf |date=2015-07-15 }} by Markus Furtner describes how to move the needle between any two locations that are parallel while sweeping negligible area. The needle will follow the shape of an "N". It moves from the first location some distance $r$ up the left of the "N", sweeps out the angle to the middle diagonal, moves down the diagonal, sweeps out the second angle,

and them moves up the parallel right side of the "N" until it reaches the required second location. The only non-zero area regions swept are the two triangles of height one and the angle at the top of the "N". The swept area is proportional to this angle which is proportional to $1/r$.

The construction starts with any triangle with height 1 and some substantial angle at the top through which the needle can easily sweep. The goal is to do many operations on this triangle to make its area smaller while keeping the directions through which the needle can sweep the same.

First, consider dividing the triangle into two and translating the pieces over each other so that their bases overlap in a way that minimizes the total area.

The needle is able to sweep out the same directions by sweeping out those given by the first triangle, jumping over to the second, and then sweeping out the directions given by the second. The needle can jump triangles using the "N" technique because the two lines at which the original triangle was cut are parallel.

Now, we divide our triangle into 2ⁿ subtriangles. The figure shows eight.

For each consecutive pair of triangles, perform the same overlapping operation we described before to get half as many new shapes, each consisting of two overlapping triangles. Next, overlap consecutive pairs of these new shapes by shifting them so that their bases overlap in a way that minimizes the total area. Repeat this n times until there is only one shape. Again, the needle is able to sweep out the same directions by sweeping those out in each of the

2ⁿ subtriangles in order of their direction. The needle can jump consecutive triangles using the "N" technique because the two lines at which these triangle were cut are parallel.

What remains is to compute the area of the final shape. Due to difficulty and length constraints the final argument cannot be fully included. Instead, an example will be shown.

Looking at the figure, it can be seen that the 2ⁿ subtriangles overlap a lot. All of them overlap at the bottom, half of them at the bottom of the left branch, a quarter of them at the bottom of the left left branch, and so on.

Suppose that the area of each shape created with i merging operations from 2ⁱ subtriangles is bounded by A_i.

Before merging two of these shapes, they have area bounded be 2A_i.

Then, move the two shapes together such that that they overlap as much as possible.

In the worst case, these two regions are

two 1 by ε rectangles perpendicular to each other so that they overlap at an area of only ε². But the two shapes that we have constructed, if long and skinny, point in much of the same direction because they are made from consecutive groups of subtriangles.

The handwaving states that they over lap by at least 1% of their area. Then the merged area would be bounded by

A_i+1 = 1.99 A_i. The area of the original triangle is bounded by 1. Hence, the area of each subtriangle is bounded by

A₀ = 2^-n and the final shape has area bounded by

A_n = 1.99ⁿ × 2^-n. In actuality, a careful summing up of all areas that do not overlap gives that the area of the final region is much bigger, namely, 1/n.

As n grows, this area shrinks to zero.

A Besicovitch set can be created by combining six rotations of a Perron tree created from an equilateral triangle.

A similar construction can be made with parallelograms.

There are other methods for constructing Besicovitch sets of measure zero aside from the 'sprouting' method.

For example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the two-dimensional plane.{{cite journal | last = Kahane | first = Jean-Pierre | author-link = Jean-Pierre Kahane | title = Trois notes sur les ensembles parfaits linéaires | journal = Enseignement Math. | volume = 15 | pages = 185–192 | year = 1969

}}

Image:KakeyaNeedleSet3.GIF

In 1941, H. J. Van Alphen{{cite journal | last = Alphen | first = H. J. | title = Uitbreiding van een stelling von Besicovitch | journal = Mathematica Zutphen B | volume = 10 | pages = 144–157 | year = 1942}} showed that there are arbitrary small Kakeya needle sets inside a circle with radius 2 + ε (arbitrary ε > 0). Simply connected Kakeya needle sets with smaller area than the deltoid were found in 1965. Melvin Bloom and I. J. Schoenberg independently presented Kakeya needle sets with areas approaching to $\backslash tfrac\{\backslash pi\}\{24\}(5\; -\; 2\backslash sqrt\{2\})$, the Bloom-Schoenberg number. Schoenberg conjectured that this number is the lower bound for the area of simply connected Kakeya needle sets. However, in 1971, F. Cunningham{{cite journal | last = Cunningham | first = F. | title = The Kakeya problem for simply connected and for star-shaped sets | journal = American Mathematical Monthly | volume = 78 | pages = 114–129 | year = 1971 | url = https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Cunningham.pdf | doi = 10.2307/2317619 | issue = 2 | jstor = 2317619 | publisher = The American Mathematical Monthly, Vol. 78, No. 2}} showed that, given ε > 0, there is a simply connected Kakeya needle set of area less than ε contained in a circle of radius 1.

Although there are Kakeya needle sets of arbitrarily small positive measure and Besicovitch sets of measure 0, there are no Kakeya needle sets of measure 0.

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 and Joshua Zahl.{{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}} 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.{{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}}{{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}}{{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}}

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

:$f\_\{*\}^\{\backslash delta\}(e)=\backslash sup\_\{a\backslash in\backslash mathbf\{R\}^\{n\}\}\backslash frac\{1\}\{m(T\_\{e\}^\{\backslash delta\}(a))\}\backslash int\_\{T\_\{e\}^\{\backslash delta\}(a)\}|f(y)|dm(y),$

where m denotes the n-dimensional Lebesgue measure. Notice that $f\_\{*\}^\{\backslash delta\}$ is defined for vectors e in the sphere Sⁿ⁻¹.

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_ε > 0 such that for any function f and all δ > 0, (see lp space for notation)

::$\backslash left\; \backslash |f\_\{*\}^\{\backslash delta\}\; \backslash right\; \backslash |\_\{L^n(\backslash mathbf\{S\}^\{n-1\})\}\; \backslash leqslant\; C\_\{\backslash epsilon\}\; \backslash delta^\{-\backslash epsilon\}\backslash |f\backslash |\_\{L^n(\backslash mathbf\{R\}^\{n\})\}.$

Some results toward proving the Kakeya conjecture are the following:

• The Kakeya conjecture is true for n = 1 (trivially) and n = 2 (Davies{{cite journal| last = Davies | first = Roy | title =Some remarks on the Kakeya problem | journal =Mathematical Proceedings of the Cambridge Philosophical Society | volume = 69 | pages = 417–421 | year = 1971 | doi = 10.1017/S0305004100046867 | issue = 3| bibcode = 1971PCPS...69..417D}}).
• In any n-dimensional space, Wolff{{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 = 651–674 | year = 1995 | issue = 3 | doi=10.4171/rmi/188| doi-access = free }} showed that the dimension of a Kakeya set must be at least (n+2)/2.
• In 2002, Katz and Tao{{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 = 231–263 | year = 2002 | doi=10.1007/BF02868476|doi-access=free| arxiv=math/0102135| s2cid=119644987}} improved Wolff's bound to $(2-\backslash sqrt\{2\})(n-4)+3$, which is better for n > 4.
• In 2000, Katz, Łaba, and Tao{{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 $\backslash mathbb\{R\}^3$|journal=The Annals of Mathematics|date=September 2000|volume=152|issue=2|pages=383– 446 |doi=10.2307/2661389|jstor=2661389|arxiv=math/0004015|s2cid=17007027}} proved that the Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2.
• In 2000, Jean Bourgain connected the Kakeya problem to arithmetic combinatoricsJ. 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.{{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 = 297–303 |date=March 2001}} which involves harmonic analysis and additive number theory.
• In 2017, Katz and Zahl{{cite journal|last1=Katz|first1=Nets Hawk|last2=Zahl|first2=Joshua|title=An improved bound on the Hausdorff dimension of Besicovitch sets in $\backslash mathbb\{R\}^3$|journal=Journal of the American Mathematical Society|date=2019|volume=32|issue=1|pages= 195– 259 |doi=10.1090/jams/907 |arxiv=1704.07210|s2cid=119322412}} improved the lower bound on the Hausdorff dimension of Besicovitch sets in 3 dimensions to $5/2+\backslash epsilon$ for an absolute constant $\backslash epsilon>0$.
• In 2025, Wang and Zahl posted on arXiv a potential proof of the Kakeya conjecture in the case n = 3.

Somewhat surprisingly, these conjectures have been shown to be connected to a number of questions in other fields, notably in harmonic analysis. For instance, in 1971, Charles Fefferman was able to use the Besicovitch set construction to show that in dimensions greater than 1, truncated Fourier integrals taken over balls centered at the origin with radii tending to infinity need not converge in L^p norm when p ≠ 2 (this is in contrast to the one-dimensional case where such truncated integrals do converge).{{cite journal

| last = Fefferman | first = Charles | author-link = Charles Fefferman

| title = The multiplier problem for the ball

| journal = Annals of Mathematics | volume = 94 | pages = 330–336 | year = 1971

| doi = 10.2307/1970864

| issue = 2

| jstor = 1970864

}}

Analogues of the Kakeya problem include considering sets containing more general shapes than lines, such as circles.

• In 1997{{cite journal

| last = Wolff | first = Thomas

| title = A Kakeya problem for circles

| journal = American Journal of Mathematics | volume = 119 | pages = 985–1026 | year = 1997

| doi = 10.1353/ajm.1997.0034|author-link =Thomas Wolff

| issue = 5

| s2cid = 120122372

}} and 1999,{{cite journal

| last1 = Wolff | first1 = Thomas

| title = On some variants of the Kakeya problem

| journal = Pacific Journal of Mathematics| volume = 190 | pages = 111–154 | year = 1999

| doi = 10.2140/pjm.1999.190.111|author-link =Thomas Wolff

| last2 = Wolff

| first2 = Thomas

| url = https://authors.library.caltech.edu/710/1/KOLpjm99.pdf| doi-access = free}} Wolff proved that sets containing a sphere of every radius must have full dimension, that is, the dimension is equal to the dimension of the space it is lying in, and proved this by proving bounds on a circular maximal function analogous to the Kakeya maximal function.

• It was conjectured that there existed sets containing a sphere around every point of measure zero. Results of Elias Stein{{cite journal | last = Stein | first = Elias

| title = Maximal functions: Spherical means

| journal =Proc. Natl. Acad. Sci. U.S.A. | volume = 73 | pages = 2174–2175 | year = 1976

| doi = 10.1073/pnas.73.7.2174

| pmid = 16592329 | issue = 7 | pmc = 430482|author-link =Elias Stein

| bibcode = 1976PNAS...73.2174S| doi-access = free

}} proved all such sets must have positive measure when n ≥ 3, and Marstrand{{cite journal

| last = Marstrand | first = J. M.

| title = Packing circles in the plane | volume = 55 | pages = 37–58 | year = 1987

| journal = Proceedings of the London Mathematical Society

| doi = 10.1112/plms/s3-55.1.37|author-link =J. M. Marstrand

}} proved the same for the case n=2.

A generalization of the Kakeya conjecture is to consider sets that contain, instead of segments of lines in every direction, but, say, portions of k-dimensional subspaces. Define an (n, k)-Besicovitch set K to be a compact set in Rⁿ containing a translate of every k-dimensional unit disk which has Lebesgue measure zero. That is, if B denotes the unit ball centered at zero, for every k-dimensional subspace P, there exists x ∈ Rⁿ such that (P ∩ B) + x ⊆ K. Hence, a (n, 1)-Besicovitch set is the standard Besicovitch set described earlier.

:The (n, k)-Besicovitch conjecture: There are no (n, k)-Besicovitch sets for k > 1.

In 1979, Marstrand{{cite journal | last = Marstrand | first = J. M.| title = Packing Planes in $\backslash mathbb\{R\}^3$ | journal = Mathematika | volume = 26 | pages = 180–183 | year = 1979|author-link =J. M. Marstrand| doi = 10.1112/S0025579300009748

| issue = 2}} proved that there were no (3, 2)-Besicovitch sets. At around the same time, however, Falconer{{cite journal | last = Falconer | first = K. J. | author-link=Kenneth Falconer (mathematician)| title = Continuity properties of k-plane integrals and Besicovitch sets | journal = Mathematical Proceedings of the Cambridge Philosophical Society | volume = 87 | pages = 221–226 | year = 1980 | doi = 10.1017/S0305004100056681| issue = 2| bibcode = 1980MPCPS..87..221F}} proved that there were no (n, k)-Besicovitch sets for 2k > n. The best bound to date is by Bourgain,{{cite journal

| last = Bourgain | first = Jean | title = Besicovitch type maximal operators and applications to Fourier analysis | journal = Geometric and Functional Analysis | volume = 1 | pages = 147–187 | year = 1997 | doi = 10.1007/BF01896376|doi-access=|author-link =Jean Bourgain

| issue = 2| s2cid = 122038469 }} who proved in that no such sets exist when 2^k−1 + k > n.

In 1999, Wolff posed the finite field analogue to the Kakeya problem, in hopes that the techniques for solving this conjecture could be carried over to the Euclidean case.

:Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ Fⁿ be a Kakeya set, i.e. for each vector y ∈ Fⁿ there exists x ∈ Fⁿ such that K contains a line {x + ty : t ∈ F}. Then the set K has size at least c_n|F|ⁿ where c_n>0 is a constant that only depends on n.

Zeev Dvir proved this conjecture in 2008, showing that the statement holds for c_n = 1/n!.{{cite journal |first=Z. |last=Dvir |title=On the size of Kakeya sets in finite fields |journal=Journal of the American Mathematical Society |volume=22 |pages=1093–1097 |year=2009 |issue=4 |doi=10.1090/S0894-0347-08-00607-3 |arxiv=0803.2336 |bibcode=2009JAMS...22.1093D |s2cid=3358826 }}{{cite web |url=http://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/ |title=Dvir's proof of the finite field Kakeya conjecture |access-date=2008-04-08 |author=Terence Tao |author-link=Terence Tao |date=2008-03-24|work=What's New}} In his proof, he observed that any polynomial in n variables of degree less than |F| vanishing on a Kakeya set must be identically zero. On the other hand, the polynomials in n variables of degree less than |F| form a vector space of dimension

:$\{|\backslash mathbf\{F\}|+n-1\backslash choose\; n\}\backslash ge\; \backslash frac\{|\backslash mathbf\{F\}|^n\}\{n!\}.$

Therefore, there is at least one non-trivial polynomial of degree less than |F| that vanishes on any given set with less than this number of points. Combining these two observations shows that Kakeya sets must have at least |F|ⁿ/n! points.

It is not clear whether the techniques will extend to proving the original Kakeya conjecture but this proof does lend credence to the original conjecture by making essentially algebraic counterexamples unlikely. Dvir has written a survey article on progress on the finite field Kakeya problem and its relationship to randomness extractors.{{Cite journal

|id={{ECCC|2009|09|077}} |title=From Randomness Extraction to Rotating Needles |year=2009|first=Zeev|last=Dvir|journal=ACM SIGACT News}}.

• Nikodym set

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| mr=2003254

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• [http://www.math.ubc.ca/~ilaba/kakeya.html Kakeya at University of British Columbia ]
• [https://www.math.ucla.edu/~tao/java/Besicovitch.html Besicovitch at UCLA]
• [http://mathworld.wolfram.com/KakeyaNeedleProblem.html Kakeya needle problem at mathworld]
• [http://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/ Dvir's proof of the finite field Kakeya conjecture at Terence Tao's blog]
• [https://www.math.stonybrook.edu/~bishop/lectures/UW.pdf An Introduction to Besicovitch-Kakeya Sets]

{{DEFAULTSORT:Kakeya Set}}

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