Besicovitch sets

A Besicovitch set is a set in the plane (or in higher dimensions) which contains at least one unit line segment in every direction. The triangle pictured below forms one quarter of a Besicovitch set, if its base and height are exactly one unit long: you can place a unit line segment in any direction between N-S and NW-SE (which forms a quarter of all possible orietnations), simply by placing one end at the top corner.

Around the turn of the century, it was thought that there was a certain minimum size to Besicovitch sets; that you could never have a Besicovitch set which had an area smaller than pi/8. (In particular, you could never have a quarter-Besicovitch set whose area was smaller than pi/32). However, this conjecture was disproved by A.S. Besicovitch, who showed that you could have a Besicovitch set of arbitrarily small area (or even zero area)!

The applet below gives one version of Besicovitch's construction. Basically the idea is to cut up the triangle you see below and shove the pieces together so that there is a lot of overlap. There are two parameters: n, which controls how many times you cut the triangle up, and alpha, which controls how much you shove things closer together. By choosing the two parameters carefully, you can make a quarter-Besicovitch set whose area is as small as you please. By putting four of these quarter-Besicovitch sets together, you can manufacture a genuine Besicovitch set with arbitrarily small area.

In this demo, n can be set to any integer from 0 to 9, and alpha can be any number from 0.5 to 1. You have to click the "Redraw" button to actually redraw the Besicovitch set. A crude upper bound for the area of the set is also given at the bottom of the applet; it is off by a factor of two or so, but I'm too lazy to find the exact area. I suggest setting alpha to 0.8 and incrementing n from 0 to 9 to get an idea of the construction.

Details of the construction (together with applications to summation of multiple Fourier series) can be found in

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