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| [[Image:Kakeya needle.gif|right|thumb|Needle shown rotating inside a [[deltoid curve|deltoid]]. At every stage of its rotation, the needle is in contact with the deltoid at three points: two endpoints (blue) and one tangent point (black).
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| The needle's midpoint (red) describes a circle with diameter equal to half the length of the needle.|208px]]
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| In [[mathematics]], a '''Kakeya set''', or '''Besicovitch set''', is a set of points in [[Euclidean space]] which contains a unit [[line segment]] in every direction. For instance, a [[Disk (mathematics)|disk]] of radius 1/2 in the [[Euclidean plane]], or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. [[Abram Samoilovitch Besicovitch|Besicovitch]] showed that there are Besicovitch sets of [[measure zero]].
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| A '''Kakeya needle set''' (sometimes also known as a Kakeya set) is a (Besicovitch) set in the plane with a stronger property, that a unit line segment can be rotated continuously through 180 degrees within it, returning to its original position with reversed orientation. Again, the disk of radius 1/2 is an example of a Kakeya needle set. Besicovitch showed that there are Kakeya needle sets of arbitrarily small positive measure.
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| ==Kakeya needle problem==
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| 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 set|convex]] regions, by {{harvs|txt|authorlink=Soichi Kakeya|first=Soichi|last= Kakeya|year=1917}}. The minimum area for convex sets is achieved by an [[equilateral triangle]] of height 1 and area 1/√3, as [[Gyula Pál|Pál]] showed.<ref>{{cite journal
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| | last = Pal | first = Julius
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| | title = Ueber ein elementares variationsproblem
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| | journal = Kongelige Danske Videnskabernes Selskab Math.-Fys. Medd.
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| | volume = 2 | pages = 1–35 | year = 1920 }}</ref>
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| Kakeya seems to have suggested that the Kakeya set ''D'' of minimum area, without the convexity restriction, would be a three-pointed [[deltoid curve|deltoid]] shape. However, this is false; there are smaller non-convex Kakeya sets.
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| ==Besicovitch sets==
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| [[Image:Perron tree.svg|thumb|A "sprouting" method for constructing a Kakeya set of small measure. Shown here are two possible ways of dividing our triangle and overlapping the pieces to get a smaller set, the first if we just use two triangles, and the second if we use eight. Notice how small the sizes of the final figures are in comparison to the original starting figure.]]
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| [[Abram Samoilovitch Besicovitch|Besicovitch]]<ref>{{cite journal | last=Besicovitch | first=Abram | authorlink=Abram Samoilovitch Besicovitch | title=Sur deux questions d'integrabilite des fonctions | journal=J. Soc. Phys. Math. | pages=105–123 | volume=2 | year=1919}}<br>{{cite journal | last=Besicovitch | first=Abram | authorlink=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}}</ref> 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 round. 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 showing such a set could have arbitrarily small [[measure (mathematics)|measure]] was from 1919. The problem may have been considered by analysts before that.
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| One method of constructing a Besicovitch set can be described as follows (see figure for corresponding illustrations). The following is known as a "Perron tree" after O. Perron who was able to simplify Besicovitch's original construction:<ref>{{cite journal | last=Perron | first=O. | title=Über eine Satz von Besicovitch | journal=Mathematische Zeitschrift | volume=28 | pages=383–386 | year=1928 | doi=10.1007/BF01181172}}<br>{{cite book | last=Falconer | first=K. J. | title=The Geometry of Fractal Sets | publisher=Cambridge University Press | year=1985 | pages=96–99}}</ref> take a triangle with height 1, divide it in two, and translate both pieces over each other so that their bases overlap on some small interval. Then this new figure will have a reduced total area.
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| Now, suppose we divide our triangle into eight subtriangles. For each consecutive pair of triangles, perform the same overlapping operation we described before to get four new shapes, each consisting of two overlapping triangles. Next, overlap consecutive pairs of these new shapes by shifting their bases over each other partially, so we're left with two shapes, and finally overlap these two in the same way. In the end, we get a shape looking somewhat like a tree, but with an area much smaller than our original triangle.
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| To construct an even smaller set, subdivide your triangle into, say, <math>2^n</math> triangles each of base length <math>2^{-n}</math>, and perform the same operations as we did before when we divided our triangle twice and eight times. If the amount of overlap we do on each triangle is small enough and the size <math>n</math> of the subdivision of our triangle is large enough, we can form a tree of area as small as we like. A Besicovitch set can be created by combining three rotations of a Perron tree created from an equilateral triangle.
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| Adapting this method further, we can construct a sequence of sets whose intersection is a Besicovitch set of measure zero. One way of doing this is to observe that if we have any parallelogram two of whose sides are on the lines ''x'' = 0 and ''x'' = 1 then we can find a union of parallelograms also with sides on these lines, whose total area is arbitrarily small and which contain translates of all lines joining a point on
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| ''x'' = 0 to a point on ''x'' = 1 that are in the original parallelogram. This follows from a slight variation of Besicovich's construction above. By repeating this we can find a sequence of sets
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| :<math> K_0\supseteq K_1 \supseteq K_2 \cdots</math>
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| each a finite union of parallelograms between the lines ''x'' = 0 and ''x'' = 1, whose areas tend to zero and each of which contains translates of all lines joining ''x'' = 0 and ''x'' = 1 in a unit square. The intersection of these sets is then a measure 0 set containing translates of all these lines, so a union of two copies of this intersection is a measure 0 Besicovich set.
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| There are other methods for constructing Besicovitch sets of measure zero aside from the 'sprouting' method. For example, [[Jean-Pierre Kahane|Kahane]]<ref>{{cite journal
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| | last = Kahane | first = Jean-Pierre | authorlink = Jean-Pierre Kahane
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| | title = Trois notes sur les ensembles parfaits linéaires
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| | journal = Enseignement Math. | volume = 15 | pages = 185–192 | year = 1969
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| }}</ref> uses [[Cantor set]]s to construct a Besicovitch set of measure zero in the two-dimensional plane.
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| [[Image:KakeyaNeedleSet3.GIF|thumb|A Kakeya needle set constructed from Perron trees.]]
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| ==Kakeya needle sets==
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| By using a trick of Pál, known as '''Pál joins''' (given two parallel lines, any unit line segment
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| can be moved continuously from one to the other on a set of arbitrary small measure), a set in which a unit line segment can be rotated continuously through 180 degrees can be created from a Besicovitch set consisting of Perron trees.<ref>[http://www.mathematik.uni-muenchen.de/~lerdos/Stud/furtner.pdf The Kakeya Problem] by Markus Furtner</ref>
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| In 1941, H. J. Van Alphen<ref>{{cite journal
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| | last = Alphen | first = H. J. | title = Uitbreiding van een stelling von Besicovitch
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| | journal = Mathematica Zutphen B | volume = 10 | pages = 144–157 | year = 1942
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| }}</ref> showed that there are arbitrary small Kakeya needle sets inside a circle with radius <math>2 + \epsilon</math> (arbitrary <math>\epsilon > 0</math>).
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| [[Simply connected]] Kakeya needle sets with smaller area than the deltoid were
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| found in 1965. Melvin Bloom and I. J. Schoenberg independently presented Kakeya needle sets with areas approaching to
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| <math>(5 - 2\sqrt{2})\pi/24</math>, the '''Bloom-Schoenberg number'''. Schoenberg conjectured that this number
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| is the lower bound for the area of simply connected Kakeya needle sets.
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| However, in 1971, F. Cunningham<ref>{{cite journal
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| | last = Cunningham | first = F. | title = The Kakeya problem for simply connected and for star-shaped sets
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| | journal = American Mathematical Monthly | volume = 78 | pages = 114–129 | year = 1971
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| | url = http://mathdl.maa.org/images/upload_library/22/Ford/Cunningham.pdf
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| | doi = 10.2307/2317619
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| | issue = 2
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| | jstor = 2317619
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| | publisher = The American Mathematical Monthly, Vol. 78, No. 2
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| }}</ref> showed that, given <math>\epsilon > 0</math>, there is a simply connected Kakeya needle set of area less than
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| <math>\epsilon</math> contained in a circle of radius 1.
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| Although there are Kakeya needle sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.
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| ==Kakeya conjecture==
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| ===Statement===
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| 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 dimension s less than the dimension of the space in which they lie? This question gives rise to the following conjecture:
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| :: '''Kakeya set conjecture''': Define a ''Besicovitch set'' in '''R'''<sup>n</sup> to be a set which contains a unit line segment in every direction. Is it true that such sets necessarily have [[Hausdorff dimension]] and [[Minkowski dimension]] equal to ''n''?
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| This is known to be true for ''n'' = 1, 2 but only partial results are known in higher dimensions.
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| ===Kakeya maximal function===
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| 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 <math>\delta>0</math>, 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
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| :<math> f_{*}^{\delta}(e)=\sup_{a\in\mathbb{R}^{n}}\frac{1}{m(T_{e}^{\delta}(a))}\int_{T_{e}^{\delta}(a)}|f(y)|dm(y)</math> | |
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| 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>.
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| Then there is a conjecture for these functions that, if true, will imply the Kakeya set conjecture for higher dimensions:
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| ::'''Kakeya maximal function conjecture''': For all <math>\epsilon>0</math>, there exists a constant <math>C_{\epsilon}>0</math> such that for any function ''f'' and all <math>\delta>0</math>,
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| <div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><math> \|f_{*}^{\delta}\|_{L^n(\mathbf{S}^{n-1})}\leqslant C_{\epsilon}\delta^{-\epsilon}\|f\|_{L^n(\mathbf{R}^{n})}. </math></div>
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| ::(see [[lp space]] for notation).
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| === Results===
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| Some results toward proving the Kakeya conjecture are the following:
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| * The Kakeya conjecture is true for ''n'' = 1 (trivially) and ''n'' = 2 (Davies<ref>{{cite journal
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| | last = Davies | first = Roy | title =Some remarks on the Kakeya problem
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| | journal =Proc. Cambridge Philos. Soc. | volume = 69 | pages = 417–421 | year = 1971
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| | doi = 10.1017/S0305004100046867
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| | issue = 3
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| }}</ref>).
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| * In any n-dimensional space, Wolff<ref>{{cite journal
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| | last = Wolff | first = Thomas | authorlink = Thomas Wolff
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| | title = An improved bound for Kakeya type maximal functions
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| | journal = Rev. Mat. Iberoamericana | volume = 11 | pages = 651–674 | year = 1995
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| }}</ref> showed that the dimension of a Kakeya set must be at least <math>\tfrac{1}{2}</math>(''n''+2).
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| * In 2002, Katz and Tao<ref>{{cite journal | last=Katz | first=Nets Hawk| coauthors=Tao, Terence
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| | title = New bounds for Kakeya problems
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| | journal = J. Anal. Math. | volume = 87 | pages = 231–263 | year = 2002 | doi=10.1007/BF02868476
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| }}</ref> improved Wolff's bound to <math>(2-\sqrt{2})(n-4)+3</math>, which is better for ''n''>4.
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| * 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
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| | last = Tao | first = Terence | authorlink = Terence Tao
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| | title = From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis and PDE| url=http://www.ams.org/notices/200103/fea-tao.pdf
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| | journal = Notices of the AMS | volume = 48 | issue = 3 | pages = 297–303 |date=March 2001}}</ref> which involves [[harmonic analysis]] and [[additive number theory]].
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| ==Applications to analysis==
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| 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]]<ref>{{cite journal
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| | last = Fefferman | first = Charles | authorlink = Charles Fefferman
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| | title = The multiplier problem for the ball
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| | journal = Annals of Mathematics | volume = 94 | pages = 330–336 | year = 1971
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| | doi = 10.2307/1970864
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| | issue = 2
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| | jstor = 1970864
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| }}</ref> 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 [[Lp space|''L''<sup>''p''</sup> norm]] when ''p'' ≠ 2 (this is in contrast to the one-dimensional case where such truncated integrals do converge).
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| ==Analogues and generalizations of the Kakeya problem==
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| ===Sets containing circles and spheres===
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| Analogues of the Kakeya problem include considering sets containing more general shapes than lines, such as circles.
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| * In 1997<ref>{{cite journal
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| | last = Wolff | first = Thomas
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| | title = A Kakeya problem for circles
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| | journal = American Journal of Mathematics | volume = 119 | pages = 985–1026 | year = 1997
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| | doi = 10.1353/ajm.1997.0034|authorlink =Thomas Wolff
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| | issue = 5
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| }}</ref> and 1999,<ref>{{cite journal
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| | last = Wolff | first = Thomas
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| | title = On some variants of the Kakeya problem
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| | journal = Pacific Journal of Mathematics| volume = 190 | pages = 111–154 | year = 1999
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| | doi = 10.2140/pjm.1999.190.111|authorlink =Thomas Wolff
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| | last2 = Wolff
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| | first2 = Thomas
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| }}</ref> 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.
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| * It was conjectured that there existed sets containing a sphere around every point of measure zero. Results of [[Elias Stein]]<ref>{{cite journal | last = Stein | first = Elias
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| | title = Maximal functions: Spherical means
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| | journal =Proc. Natl. Acad. Sci. U.S.A. | volume = 73 | pages = 2174–2175 | year = 1976
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| | doi = 10.1073/pnas.73.7.2174
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| | pmid = 16592329 | issue = 7 | pmc = 430482|authorlink =Elias Stein
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| }}</ref> proved all such sets must have positive measure when <math>n\geq 3</math>, and Marstrand<ref>{{cite journal
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| | last = Marstrand | first = J. M.
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| | title = Packing circles in the plane | volume = 55 | pages = 37–58 | year = 1987
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| | journal = Proceedings of the London Mathematical Society
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| | doi = 10.1112/plms/s3-55.1.37|authorlink =J. M. Marstrand
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| }}</ref> proved the same for the case ''n=2''.
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| ===Sets containing k-dimensional disks===
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| 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 '''R'''<sup>n</sup> 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'' ∈ '''R'''<sup>n</sup> such that <math>(P\cap B)+x\subseteq K</math>. Hence, a ''(n,1)''-Besicovitch set is the standard Besicovitch set described earlier.
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| ::'''The (n,k)-Besicovitch conjecture:''' There are no ''(n,k)''-Besicovitch sets for ''k>1''.
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| In 1979, Marstrand<ref>{{cite journal
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| | last = Marstrand | first = J. M.
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| | title = Packing Planes in R<sup>3</sup>
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| | journal = Mathematika | volume = 26 | pages = 180–183 | year = 1979|authorlink =J. M. Marstrand
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| | doi = 10.1112/S0025579300009748
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| | issue = 2
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| }}</ref> proved that there were no ''(3,2)''-Besicovitch sets. At around the same time, however, [[Kenneth Falconer (mathematician)|Falconer]]<ref>{{cite journal
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| | last = Falconer | first = K. J. | authorlink=Kenneth Falconer (mathematician)
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| | title = Continuity properties of k-plane integrals and Besicovitch sets
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| | journal = Math. Proc. Cambridge Philos. Soc. | volume = 87 | pages = 221–226 | year = 1980
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| | doi = 10.1017/S0305004100056681|authorlink =K. J. Falconer
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| | issue = 2
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| }}</ref> proved that there were no ''(n,k)''-Besicovitch sets for ''2k>n''. The best bound to date is by Bourgain,<ref>{{cite journal
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| | last = Bourgain | first = Jean
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| | title = Besicovitch type maximal operators and applications to Fourier analysis
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| | journal = Geom. Funct. Anal. | volume = 1 | pages = 147–187 | year = 1997
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| | doi = 10.1007/BF01896376|authorlink =Jean Bourgain
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| | issue = 2
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| }}</ref> who proved in that no such sets exist when 2<sup>k-1</sup>+''k>n''.
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| ===Kakeya sets in vector spaces over finite fields===
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| 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.
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| ::'''Finite Field Kakeya Conjecture''': Let '''F''' be a finite field, let ''K'' ⊆ '''F'''<sup>n</sup> be a Kakeya set, i.e. for each vector ''y'' ∈ '''F'''<sup>n</sup> there exists ''x'' ∈ '''F'''<sup>n</sup> such that ''K'' contains a line {''x''+''ty'': ''t'' ∈ '''F'''} . Then the set ''K'' has size at least ''c''<sub>''n''</sub>|'''F'''|<sup>n</sup> where ''c''<sub>''n''</sub>>0 is a constant that only depends on ''n''.
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| Zeev Dvir<ref>Z. Dvir, On the size of Kakeya sets in finite fields.
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| J. Amer. Math. Soc., 22:1093-1097, 2009.</ref><ref>{{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 |accessdate=2008-04-08 |author=[[Terence Tao]]|date=2008-03-24|work=What's New}}</ref>
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| proved this conjecture for ''c''<sub>''n''</sub> = 1/''n''!, using the following argument. Dvir 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
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| :<math>{|\mathbf{F}|+n-1\choose n}\ge \frac{|\mathbf{F}|^n}{n!}.</math>
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| 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 Kayeka sets must have at least |'''F'''|<sup>''n''</sup>/''n''! points.
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| 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.
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| Dvir has written a survey article on recent (as of 2009) progress on the finite field Kakeya problem and its relationship to [[randomness extractor]]s.<ref>{{Cite journal
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| |id={{ECCC|2009|09|077}}
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| |title=From Randomness Extraction to Rotating Needles
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| |year=2009
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| |first=Zeev|last=Dvir
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| }}.</ref>
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| ==See also==
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| * [[Nikodym set]]
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| ==Notes==
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| {{reflist|colwidth=30em}}
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| ==References==
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| *{{cite journal
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| | last = Besicovitch | first = Abram | authorlink = Abram Samoilovitch Besicovitch
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| | title = The Kakeya Problem
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| | journal = American Mathematical Monthly | volume = 70 | pages = 697–706 | year = 1963
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| | doi = 10.2307/2312249
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| | issue = 7
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| | jstor = 2312249
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| }}
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| *{{Cite journal | last1=Dvir | first1=Zeev | title=On the size of Kakeya sets in finite fields | arxiv=0803.2336 | doi=10.1090/S0894-0347-08-00607-3 | mr=2525780 | year=2009 | journal=[[Journal of the American Mathematical Society]] | volume=22 | issue=4 | pages=1093–1097}}
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| * {{cite book
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| | last = Falconer | first = K. J.
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| | title = The Geometry of Fractal Sets
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| | publisher = Cambridge University Press | year = 1985}}
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| *{{Cite journal | last1=Kakeya | first1=Soichi | title=Some problems on maximum and minimum regarding ovals | year=1917 | journal=Tohoku science reports | volume=6 | pages=71–88}}
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| *{{cite journal
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| | last1 = Katz | first1 = Nets Hawk | author1-link = Nets Katz
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| | last2 = Łaba | first2 = Izabella | author2-link = Izabella Łaba
| |
| | last3 = Tao | first3 = Terence | author3-link = Terence Tao
| |
| | title = An improved bound on the Minkowski dimension of Besicovitch sets in R<sup>3</sup>
| |
| | journal = Annals of Mathematics | volume = 152 | pages = 383–446 | year = 2000
| |
| | url = http://www.emis.de/journals/Annals/152_2/laba.pdf
| |
| | doi = 10.2307/2661389
| |
| | issue = 2
| |
| | jstor = 2661389
| |
| }}
| |
| | |
| * {{cite book
| |
| | last = Wolff | first = Thomas | authorlink = Thomas Wolff
| |
| | chapter = Recent work connected with the Kakeya problem
| |
| | title = Prospects in Mathematics | editor = H. Rossi (ed.)
| |
| | publisher = AMS | year = 1999}}
| |
| * {{cite book
| |
| | last = Wolff | first = Thomas | authorlink = Thomas Wolff
| |
| | title = Lectures in Harmonic Analysis
| |
| | publisher = AMS | year = 2003}}
| |
| | |
| ==External links==
| |
| *[http://www.math.ubc.ca/~ilaba/kakeya.html Kakeya at University of British Columbia ]
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| *[http://www.math.ucla.edu/~tao/java/Besicovitch.html Besicovitch at UCLA]
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| *[http://mathworld.wolfram.com/KakeyaNeedleProblem.html Kakeya needle problem at mathworld]
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| *[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]
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| | |
| {{DEFAULTSORT:Kakeya Set}}
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| [[Category:Harmonic analysis]]
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| [[Category:Real analysis]]
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| [[Category:Discrete geometry]]
| |