I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6):
Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume WLOG $q > 0$.
Define the sequence $(\xi_n)$ by $\xi_n = <n\theta>$ where $<>$ denotes the fractional part. Show that this sequence is equidistributed on the points of the form $0, \frac{1}{q},..., \frac{q-1}{q}$
In fact, show that $\forall 0 \leq a < q$,
$\frac{\operatorname{card}\{n: 1 \leq n \leq N, <n\theta> = \frac{a}{q}\}}{N} = \frac{1}{q} + O(\frac{1}{N})$
($\operatorname{card}$ is cardinality)
My problem is that I don't know what I should be doing. The course is Fourier analysis, and this topic (Weyl's equidistribution theorem) is covered in only a few pages in this chapter. I don't see how I can work with the definition of equidistribuition: for every $(a,b) \subset [0,1)$, $(\xi_n)$ is equidistributed in $[0,1)$ if $\operatorname{lim}_{N \to \infty} \frac{\operatorname{card}\{n: 1 \leq n \leq N, \xi_n \in (a,b) \}}{N} = b-a$
I don't have much intuition to work with; I was able to prove that the sequence $(<n\phi>)$ where $\phi$ is the golden ratio was not equidistributed in $[0,1)$ by showing that it tended to either $0$ or $1$, so I see how a sequence might not be equidistributed and how I might be able to show it.
Proving equidistributivity seems far more difficult. Could anyone help me understand how I should work with this definition and steer me into the right direction for the proof?
This question seems very algebraic to me, no Fourier theory necessary. The key observation (which you should prove!) is that if $\langle n\frac{p}{q}\rangle = \frac{a}{q}$, then $np\equiv a\pmod{q}$. Therefore, for any $n\geq 1$, the value of $a = 0,\ldots, q-1$ such that $n\frac{p}{q} = \frac{a}{q}$ corresponds exactly to the residue class of $np$ modulo $q$.
Since $p$ and $q$ are relatively prime, $p$ is a generator of $\mathbb{Z}/q\mathbb{Z}$. It follows that the sequence of residue classes $\{[np]\}_{n=1}^\infty$ in $\mathbb{Z}/q\mathbb{Z}$ takes on all of the possible residue classes of $\mathbb{Z}/q\mathbb{Z}$, and in a cyclic manner.
Perhaps with this in mind, you can finish the exercise? I hope this helped!