Kakeya set

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).The needle's midpoint (red) describes a circle with diameter equal to half the length of the needle.

In mathematics, a Kakeya set, or Besicovitch set, is any set of points in Euclidean space which contains a unit line segment in every direction. While many types of objects satisfy this property, several interesting results and questions are motivated by considering how small such sets can be.

Kakeya needle problem

The Kakeya needle problem asks whether there is a minimum area of a region "D" in the plane, in which a needle can be turned through 360°. This question was first posed, for convex regions, in 1917 by Soichi Kakeya (1886–1947), a Japanese mathematician who worked mainly in mathematical analysis.

He seems to have suggested that "D" of minimum area, without the convexity restriction, would be a three-pointed deltoid shape. The original problem was solved by Pal. [] showed that the dimension of a Kakeya set must be at least $frac\{n+2\}\{2\}$.

* In 2002, Katz and Tao [cite journal | last=Tao | first=Terence| authorlink = Terence Tao
title = New bounds for Kakeya problems
journal = J. Anal. Math. | volume = 87 | pages = 231–263 | year = 2002] improved this bound to $(2-sqrt\{2\})(n-4)+3$, which is better for n>4.

Applications to analysis

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 [cite journal
last = Fefferman | first = Charles | authorlink = Charles Fefferman
title = The multiplier problem for the ball
journal = Ann. of Math. | volume = 94 | pages = 330–336 | year = 1971
doi = 10.2307/1970864] was able to use the Besicovitch set construction to show that Fourier transform in dimensions greater than 2, when integral is over balls centered at the origin whose radius tend to infinity, does not necessarily converge in the "L"^"p" norm when "p" ≠ 2.

Analogues and generalizations of the Kakeya problem

ets containing circles and spheres

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
unused_data = |authorlink Thomas Wolff] and 1999, [cite journal
last = Wolff | first = Thomas
title = On some variants of the Kakeya problem
journal = Pacific Journal of Mathematics| volume = 190 | pages = 111–154 | year = 1999
unused_data = |authorlink Thomas Wolff] 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. USA | volume = 73 | pages = 2174–2175 | year = 1976
doi = 10.1073/pnas.73.7.2174
pmid = 16592329 | unused_data = |authorlink Elias Stein] proved all such sets must have positive measure when $ngeq\; 3$, and Marstrand [cite journal
last = Marstrand | first = J. M.
title = Packing circles in the plane | volume = 55 | pages = 37–58 | year = 1987
journal = Proc. London. Math. Soc.
doi = 10.1112/plms/s3-55.1.37
unused_data = |authorlink J. M. Marstrand] proved the same for the case "n=2".

ets containing k-dimensional disks

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 $mathbb\{R\}^\{n\}$ 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 $xinmathbb\{R\}^\{n\}$ such that $(Pcap\; B)+xsubseteq\; K$. 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 R³
journal = Mathematika | volume = 26 | pages = 180–183 | year = 1979
unused_data = |authorlink J. M. Marstrand] proved that there were no "(3,2)"-Besicovitch sets. At around the same time, however, Falconer [cite journal
last = Falconer | first = K. J.
title = Continuity properties of k-plane integrals and Besicovitch sets
journal = Math. Proc. Cambridge Philos. Soc. | volume = 87 | pages = 221–226 | year = 1980
unused_data = |authorlink K. J. Falconer] proved that there were no "(n,k)"-Besicovitch sets for $k>frac\{n\}\{2\}$. 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 = Geom. Funct. Anal. | volume = 1 | pages = 147–187 | year = 1997
doi = 10.1007/BF01896376
unused_data = |authorlink Jean Bourgain] who proved in that no such sets exist when $2^\{k-1\}+k>n$.

Kakeya sets in vector spaces over finite fields

In 1999, Wolff posed the finite field analogue to the Kakeya problem, in hopes that the techniques for solving this simpler conjecture could be carried over to the Euclidean case. :Finite Field Kakeya Conjecture: Let $F$ be a finite field, let $Ksubseteq\; F^\{n\}$ be a Kakeya set, i.e. for each vector $yin\; F^\{n\}$ $K$ contains a line $\{x+ty:tin\; F\}$ for some $xin\; F$. Then the set $K$ has size at least $c\_\{n\}|F|^\{n\}$ where $c\_\{n\}>0$ is a constant that only depends on $n$.

This conjecture was proved in 2008 by Zeev Dvir (cite web|url=http://arxiv.org/abs/0803.2336|title=On the size of Kakeya sets in finite fields) using what Terence Tao called a "beautifully simple argument". 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 ] 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.

ee also

* Nikodym set

External links

* [http://www.math.ubc.ca/~ilaba/kakeya.html Kakeya at University of British Columbia ]
* [http://www.math.ucla.edu/~tao/java/Besicovitch.html Besicovitch at UCLA]
* [http://mathworld.wolfram.com/KakeyaNeedleProblem.html Kakeya needle problem at mathworld]

Notes

8. Katz, Nets Hawk; Laba, Izabella; Tao, Terence An improved bound on the Minkowski dimension of Besicovitch sets in $Rsp 3$. Ann. of Math. (2) 152 (2000), no. 2, 383--446

References

*cite journal
last = Besicovitch | first = Abram | authorlink = Abram Samoilovitch Besicovitch
title = The Kakeya Problem
journal = American Mathematical Monthly | volume = 70 | pages = 697–706 | year = 1963
doi = 10.2307/2312249
* cite book
last = Falconer | first = K. J.
title = The Geometry of Fractal Sets
publisher = Cambridge University Press | year = 1985
*cite journal
last = Tao | first = Terence | authorlink = Terence Tao
title = From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis and PDE
journal = Notices of the AMS | volume = 48 | issue = 3 | pages = 297–303 | year=2001 | month=March

* 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