# Applications of equidistribution

What are applications of equidistributed sequences ?

I'm looking for examples of problems/questions in fields (not necessarily mathematical) where equidistribution is an ad-hoc tool towards a solution.

• Does this is what you want (one of them)? Also, you can get a look at Achille hui's answer here. – Krokop Dec 31 '14 at 10:29

Before I list some applications of equidistribution, I need to maek clear what equidistribution is and is not. Especially what it is, because many people have a rather narrow view of it.

First of all, equidistribution is neither a sub-area of number theory or analysis nor a technique. It is a principle and as such it permeates every field of mathematics to some extent (as we will see, this includes some hardcore algebra).

Secondly, equidistribution is not a game played only on $[0,1]$ or the real numbers. Kuipers and Niederreiter's classical text considers equidistribution in general topological spaces and groups.

Thirdly, sequences are not the only thing that can be equidistributed. Any system involving a parameter is amenable to the notion, whether the index set is the integers, the reals, a real or complex vector space or a suitable index category. All you need is a directed object, a notion of averaging on the target and some completeness.

Fourthly, when experts speak of equidistribution they do not necessarily mean a limiting distribution that gives a fair share of the sequence to every corner of the space, although that is what the name implies and it is the most important limiting distribution. What the experts mean is that the sequence does have a limiting distribution. In fact, equidistribution theory should be better named 'limiting distribution theory'.

Finally, finding a limiting distribution is not the end of the investigation, it is a little below the midpoint. A limiting distribution is not good enough for practical reasons, you want to know what happens before we reach the limit; this is where the crucial notion of discrepancy comes in, and that can get pretty complicated if your host space has rich geometry: then you have to define lots of discrepancies adapted to various shapes in your space, and because discrepancy is not a limiting statement it is very delicate with respect to the shape. I am mentioning this because discrepancy estimates is one of the most applicable aspects of equidistribution theory.

Now on to the applications.

As a principle, since equidistribution is a limit of counting processes, naturally it helps simplify difficult counting problems by passing to the limit. For example:

1. Counting number of closed geodesics of some bounded length on a hyperbolic surface, or more generally totally geodesic tori in symmetric spaces. See http://www.math.harvard.edu/~ctm/papers/home/text/papers/mixing/mixing.pdf for an approachable introduction.

2. Counting prime numbers; the prime number theorem is an equidistribution theorem, and the Riemann Hypothesis an optimal discrepancy bound. By bounding this discrepancy optimally, you guarantee that there are no notable gaps in the distribution of prime numbers that could be exploited to optimize brute force factorization (for cracking public key cryptosystems) by avoiding 'gappy' regions of the search space.

3. Proving density of a sequence in a space. This is a paradoxical but very common technique: you have a sequence in a space that you want to prove is dense. Prove the stronger statement that it is equidistributed and you are done.

4. Counting and more generally the distribution of rational points on algebraic groups, their homogeneous spaces and algebraic varieties. Algebraic varieties over a field $k$ can have many points in an algebraic closure, but how do we know about the existence of points in $k$ or finite extensions? For a large class of varieties this question is one of equidistribution (by the paradoxical scheme equidistribution implies density). See Yuri Tchinkel's amazing exposition here: http://www.cims.nyu.edu/~tschinke/papers/yuri/08cmi/cmi4.pdf

5. An amazing instance of the previous theme is found in conjunction with Weil's conjectures. In proving the Weil conjectures, Deligne formulated his equidistribution theorem linking in a fascinating way projective varieties and affine (compact real) groups; this formed part of the inspiration for Katz and Sarnak's proof that the statistics of zeros of zeta functions of algebraic varieties come from random matrix (Haar) ensembles on appropriate compact groups or symmetric spaces of compact type. The entire proof is found in Katz & Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy.

6. Manjul Bhargava's groundbreaking work on the number of low-degree field extensions of bounded discriminant is based on a counting problem that involves equidistribution techniques. This is one case (of the many) where the object being equidistributed is an algebraic structure! For a flavor, see the paper http://arxiv.org/pdf/1309.2025 or dive directly into the sequence of Bhargava's Annals papers (the original based on his thesis is not too hard to read due to his clear writing; it gets more and more involved with each successive paper though). On the very first page of http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n2-p10.pdf you have an equidistribution statement :)

7. For a more applied flavor, numerical integration and the entire field of approximation theory hinge on finding discrete sequences in a space that are distributed well enough to replace integration by weighted summation on the sequence. The basic results that makes this possible is the Koksma-Hlawka inequality which states, roughly, that the difference between the integral and a summation over a sample is bounded by the discrepancy of the sample times the variation of the integrand. This has been extended to a vast number of settings and is one of the most useful mathematical statements outside of mathematics.

8. In fact, the best place to look for applications of equidistribution outside mathematics is everywhere. That is because equidistribution goes hand in hand with the ubiquitous sampling problem. Sampling by independent trials is not a good idea, because independence guarantees that you will have long strings of bias, and you do not know where they will be. What you want is low discrepancy samples and that is the purview of equidistribution.

9. Physicists care about distribution of eiqenfunctions of Hamiltonians on manifolds as the primal example of quantum chaos. The mathematical statement is called quantum ergodicity and is essentially an equidistribution statement, see the wiki article http://en.wikipedia.org/wiki/Quantum_ergodicity and Sarnak's book Arithmetic Quantum Chaos for a splendid introduction (not only to the arithmetic theory, it also has a good exposition on the general physical problem!). The book is freely available online!

There are so many more applications outside of mathematics that this post would never need to end. However, I need to stop somewhere so I will let you follow the links above and branch out. The starting point for the theory is Kuipers and Niederreiter Uniform Distribution of Sequences, but as I have indicated above, there is hardly a mathematical field that does not exploit the principle of equidistribution. For those that don't, well, what are you waiting for?

EDIT: Oh dear, I forgot to mention the overwhelming number of applications of equidistribution in Computer Science. Because I am too tired for a big edit, if you are a computer scientist and want to see such applications, I wholeheartedly recommend Chazelle's book The Discrepancy Method. WARNING: some of the proofs there are WRONG and some of the exercises undoable. However, the prose and topic selection makes for an excellent introduction to discrepancy theory in CS and related fields!

This is not too broad, but it reminds me of a solution to a specific problem:

Show that the sequence $$tan(n) ; n=1,2,3,...$$ diverges.

We (can) show this by using equidistribution of $$\pi$$ ( and therefore $$\pi/2) mod1$$, meaning the decimal parts of $$\{\pi, 2\pi,..., n\pi,...\}$$ are dense in the unit interval. It follows that the sequence $$a_n =n$$ will be indefinitely close to (but never actually equal to, by irrationality of $$\pi$$) to $$(4k \pm 1)\pi/2$$ , so that $$tan(n)$$ will blow up for infinitely-many n, meaning beyond any specific/fixed value of n. So $$tan(n)$$ diverges. EDIT: What I mean here is that the decimal parts of $$\{\pi,2pi,..., n\pi,...\}$$ are dense on the unit interval. I stand by it despite the downvote.

• What in the world might it mean to say that $\pi$ is equidistributed? $\pi$ is not a sequence. – David C. Ullrich Jun 26 '19 at 21:35
• I meant that the sequence {\pi, 2\pi,..., n\pi,.....} is dense in [0,1] mod1 , (meaning the decimal parts) because \pi is irrational; in that sense, \pi is equidistributed mod1. I think this is a corollary of the Ergodic theorem. – MSIS Jun 26 '19 at 21:38
• Yes, that sequence is dense. Saying it's dense does not mean it's equidistributed! Although it is equidistributed. In any case it would be good iif you said what you meant instead of saying that $\pi$ is equidistributed; that's nonsense. – David C. Ullrich Jun 26 '19 at 21:52
• Maybe I stated in a confusing way but I never equated dense with equidistributed. At least I did not mean to. – MSIS Jun 26 '19 at 21:58
• Your first comment above reads "...is dense...; in that sense, ... is equidistributed". – David C. Ullrich Jun 26 '19 at 22:01