Weyl's Criterion for Equidistribution
1. Definition and Statement
Definition of Equidistribution Suppose that \(\{x_n\}_{n=1}^\infty\) is a sequence of real numbers in \([0,1]\). We say that the sequence is equidistributed when for any real numbers \(a < b\) in \([0,1]\),
\[{}\lim_{N \rightarrow \infty} \frac{\# \{n \ : \ 1 \leq n \leq N \text{ and } x_n \in [a,b]\}}{N}{}\]
\[{}= b-a.{}\]
Proving that a sequence is equidistributed is no small task. The first key observation is to note that saying that a sequence \(\{x_n\}_{n=1}^\infty \subset [0,1]\) is equidistributed is in fact equivalent to the assertion that the formula
holds for all \(f\) of the form \(f = \chi_{[a,b]}\) (i.e., the indicator function of \([a,b]\)) for \([a,b] \subset [0,1]\). This is because we may write
\[{}\# \left\{n \ : \ 1 \leq n \leq N \text{ and } x_n \in [a,b] \right\}{}\]
\[{}= \sum_{n=1}^N \chi_{[a,b]}(x_n){}\]
(and because \(b-a = \int_0^1 \chi_{[a,b]}(t) dt\)). It's also important to observe that for any fixed sequence \(\{x_n\}_{n=1}^\infty\), the class of Riemann-integrable functions satisfying (1) is a vector space (because both sides are linear in \(f\)).
Theorem (Weyl's Criterion)
The sequence \(\{x_n\}_{n=1}^\infty\) is equidistributed in \([0,1]\) if and only if for all integers \(\ell \neq 0\),
\[ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e^{2 \pi i \ell x_n} = 0. \]
Proof
Corollary
Let \(\alpha\) be any real number and let \(x_n\) be the fractional part of \(n \alpha\) for each \(n\). This sequence is equidistributed if and only if \(\alpha\) is irrational.
Proof
The fractional part \(\left<x\right>\) of a real number \(x\) is simply \(x\) minus the largest integer not exceeding \(x\), i.e., \(x - \lfloor x \rfloor\). Because \(e^{2 \pi i \ell x}\) is 1-periodic and because \(\lfloor x \rfloor \in {\mathbb Z}\), it must always be the case that \(e^{2 \pi i \ell \left<x\right>} = e^{2 \pi i \ell x}\) for all real \(x\). This means that
\[ \sum_{n=1}^N e^{2 \pi i \ell \left<n \alpha\right>} = \sum_{n=1}^N e^{2 \pi i n \ell \alpha}. \]
The right-hand side is a finite geometric series whose sum we can compute explicitly:
\[{}\frac{1}{N} \sum_{n=1}^N e^{2 \pi i \ell \left<n \alpha\right>}{}\]
\[{}= \frac{e^{2 \pi i \ell \alpha}}{N} \frac{e^{2 \pi i N \ell \alpha}-1}{e^{2 \pi i \ell \alpha - 1}}{}\]
provided that \(e^{2 \pi i \ell \alpha}\) is not equal to one (which could only happen if \(\ell \alpha\) is an integer, meaning that it only occurs when \(\ell = 0\) or when \(\alpha\) is rational). We see that the numerator remains bounded as a function of \(N\), so the expression tends to zero as \(N \rightarrow \infty\).