Functorial way, equidistribution

I am almost always surprised and excited to find functorial changes and dualities turning the problems that look hard in one category into something manageable.

Here is a puzzle that I came across recently. It’s called the rectangle tiling theorem and says that any rectangle that is tiled with rectangles with at least one integer side should also have an integral side. The result is, in fact, true if we replace integers by any subgroup of reals.

There are several ways to prove the result for the integer case. I along with my friend, tried to come up with a solution by studying the structure of the problem. The first object is to understand the phenomenon of tiling. What are the restrictions imposed by the tiles on the big rectangle and how the fact that they are all rectangles affect the tiling process?  As we started thinking, we needed a way to capture tiling first and then a way to capture integer( group) sized rectangles.  One naive idea is that the rectangle is somehow made of smaller rectangles. Thus we were attempting to find the right additive/group type structure in the situation. We realized that we can think of the rectangles as oriented loops and then the smaller loops add up to the bigger loop. Yes! we are heading somewhere. Then, the problem is to show that if every rectangle has the property of having an integer side, so does the bigger rectangle. So we tried to prove some subgroup type of characterization. We were stuck here for some time. Meanwhile, my friend Kant realized that we can try to find functions (forms if you want ) such that the integral around a rectangle vanishes if there is an integer side to it.  After some trial and error, he was able to find some function involving sine and cosine which has this property. Hence the integral over the bigger rectangle which equals the sum of integral over smaller loops is zero, which proves that it also has an integral side. Thus a task that seemed difficult when we are trying to see just in terms of the loops reduced to some easy calculations when we changed our focus to functions on these loops!. So to prove the structure of these rectangles, we did it by bringing functions on them!

Anyway if you want to look at some interesting list of proofs of the result. Here is the link.

As I was writing this, I remembered another such instance where we use functions to prove results about the space. This is Weyl’s equidistribution criterion.

is equidistributed iff for every integer , we have

This basically shows a way to prove the fact that a certain sequence of numbers is equidistributed modulo 1 by showing some exponential sums are small.

Think of equidistribution modulo 1  as giving the distribution of points on a circle and the points as defining a discrete measure on the unit circle. Then, as becomes larger the sequences of discrete measure converge to the uniform measure of the circle. Now using the integrals of functions over these measures it is enough to prove that for any function the sequences of integrals converges to . So, it is enough to prove this fact for an orthonormal basis because of linearity. Now use an exponential basis for to get the desired criterion.

So, to study some objects, studying functorial/dual and other related objects can be very helpful. I have learned to appreciate this fact more and more.