Editing Kakeya set (section)

==Besicovitch needle sets==
[[Image:Perron tree.svg|thumb|"Sprouting the Perron tree": a method for constructing a Kakeya set of small measure. Shown here are two possible ways of dividing a triangle and overlapping the pieces to get a smaller set, the first with two triangles, and the second with eight. The method can be used to construct an arbitrarily small set by cutting up the original triangle to <math>2^n</math> pieces. See <ref name=":0">{{Cite journal |last=Besicovitch |first=A. S. |date=August 1963 |title=The Kakeya Problem |url=http://dx.doi.org/10.2307/2312249 |journal=The American Mathematical Monthly |volume=70 |issue=7 |pages=697–706 |doi=10.2307/2312249 |jstor=2312249 |issn=0002-9890}}</ref> for details.]]

[[Abram Samoilovitch Besicovitch|Besicovitch]] was able to show that there is no lower bound > 0 for the area of such a region <math>D</math>, in which a needle of unit length can be turned around. That is, for every <math>\varepsilon>0</math>, there is region of area <math>\varepsilon</math> within which the needle can move through a continuous motion that rotates it a full 360 degrees.<ref>{{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}}<br>{{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 }}</ref> 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 (mathematics)|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.<ref>{{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 }}<br>{{cite book | last=Falconer | first=K. J. | title=The Geometry of Fractal Sets | publisher=Cambridge University Press | year=1985 | pages=96–99}}</ref> The precise construction and numerical bounds are given in Besicovitch's popularization.<ref name=":0" />

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 [[Gyula Pál|Pál]], known as '''Pál joins''',<ref>[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</ref> 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 <math>r</math> 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 <math>1/r</math>.

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 2<sup>''n''</sup> 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 
2<sup>''n''</sup> 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 2<sup>''n''</sup> 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 2<sup>''i''</sup> subtriangles is bounded by ''A''<sub>''i''</sub>.
Before merging two of these shapes, they have area bounded be 2''A''<sub>''i''</sub>. 
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 ε<sup>''2''</sup>. 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''<sub>''i+1''</sub> = 1.99 ''A''<sub>''i''</sub>. The area of the original triangle is bounded by 1. Hence, the area of each subtriangle is bounded by 
''A''<sub>''0''</sub> = 2<sup>''-n''</sup> and the final shape has area bounded by 
''A''<sub>''n''</sub> = 1.99<sup>''n''</sup> × 2<sup>''-n''</sup>. 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, [[Jean-Pierre Kahane|Kahane]] uses [[Cantor set]]s to construct a Besicovitch set of measure zero in the two-dimensional plane.<ref>{{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
}}</ref>

[[Image:KakeyaNeedleSet3.GIF|thumb|A Kakeya needle set constructed from Perron trees.]]

In 1941, H. J. Van Alphen<ref>{{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}}</ref> 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 [[Isaac Jacob Schoenberg|I. J. Schoenberg]] independently presented Kakeya needle sets with areas approaching to <math>\tfrac{\pi}{24}(5 - 2\sqrt{2})</math>, 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<ref>{{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}}</ref> 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.