Kakeya Set - Besicovitch Sets

Besicovitch Sets

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 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 was from 1919. The problem may have been considered by analysts before that.

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: 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.

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.

To construct an even smaller set, subdivide your triangle into, say, triangles each of base length, 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 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.

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 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

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.

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.