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.
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:


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.
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.
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.
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
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 &amp; Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy.
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 :)
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.
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.
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.