Kakeya set
[3] This built on earlier work of his, on plane sets which contain a unit segment in each orientation.
Besicovitch's work showing such a set could have arbitrarily small measure was from 1919.
[2] The first observation to make is that the needle can move in a straight line as far as it wants without sweeping any area.
The only non-zero area regions swept are the two triangles of height one and the angle at the top of the "N".
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 though which the needle can sweep the same.
First consider dividing the triangle in two and translating the pieces over each other so that their bases overlap in a way that minimizes the total area.
Suppose that the area of each shape created with i merging operations from 2i subtriangles is bounded by Ai.
In a 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.
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
Schoenberg conjectured that this number is the lower bound for the area of simply connected Kakeya needle sets.
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.
This question gives rise to the following conjecture: This is known to be true for n = 1, 2 but only partial results are known in higher dimensions.
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.
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.
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 Lp norm when p ≠ 2 (this is in contrast to the one-dimensional case where such truncated integrals do converge).
[16] Analogues of the Kakeya problem include considering sets containing more general shapes than lines, such as circles.
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.
At around the same time, however, Falconer[22] proved that there were no (n, k)-Besicovitch sets for 2k > n. The best bound to date is by Bourgain,[23] who proved in that no such sets exist when 2k−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.
Zeev Dvir proved this conjecture in 2008, showing that the statement holds for cn = 1/n!.
[24][25] 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.
Dvir has written a survey article on progress on the finite field Kakeya problem and its relationship to randomness extractors.
