If you take the fractional parts of a set of numbers **{ cos nx : integer n > 0}**, the result is uniformly distributed for almost all

**. That is, in the limit, the number of times the sequence visits a subinterval of**

*x***[0, 1]**is proportional to the length of the interval. (Clearly it's not true for all

**: take**

*x***, for instance. Or any rational number times**

*x*= 0**π**.)

The proof requires some delicate work with Fourier analysis that I'll not repeat here. If you're interested in the proof, see Theorem 4.4 of Uniform Distribution of Sequences.

This is a surprising result. Why should such a strange sequence be uniformly distributed? Let's look at a histogram to see whether the theorem is plausible.

OK. Looks plausible.

But there's a generalization that's even more surprising. Let ** a** be any increasing sequence of integers. Then the fractional parts of

*a***cos**

**are uniformly distributed for almost all**

*ax***.**

*x*Any increasing sequence of integers? Like the prime numbers, for example? Apparently so. The result produces a very similar histogram.

But this can't hold for just any sequence. Surely, you could pick an integer sequence to thwart the theorem. Pick an ** x**, then let

**be the subset of the integers for which**

*a***. Then**

*n*cos*nx*< 0.5**isn't uniformly distributed because it never visits the right half the unit interval!**

*a*cos*ax*Where's the contradiction? The theorem doesn't start by saying "For almost all ** x**..." It starts by saying "For any increasing sequence

**..." — that is, you don't get to pick**

*a***first. You pick the sequence first, then you get a statement that is true for almost all**

*x***. The theorem is true for every increasing sequence of integers, but the exceptional set of**

*x***s may be different for each integer sequence.**

*x*
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