Weyl's Criterion for Equidistribution

It's easier to start by showing that \(\{x_n\}_{n=1}^\infty\) is equidistributed if and only if (1) holds for all continuous 1-periodic functions \(f\).

The key observation is a sort of Squeeze Theorem: Suppose \(\{g_j\}_{j=1}^\infty\) and \(\{h_j\}\) are sequences of Riemann-integrable functions on \([0,1]\) satisfying (1) for each \(j\) and having \(\int_0^1 (h_j(t)-g_j(t)) dt \rightarrow 0\) as \(j \rightarrow \infty\). If \(f\) is any function such that \(g_j(x) \leq f(x) \leq h_j(x)\) for each \(j\) and all \(x \in [0,1]\), then \(f\) is Riemann integrable and satisfies (1) as well. To see this, fix any single value of \(j\) large enough so that \(\int_0^1 (h_j(t)-g_j(t)) dt < \frac{\epsilon}{2}\) and then consider all \(N\) large enough that

and likewise for \(h_j\). Then for all such \(N\),

and

Another key observation is that \(\chi_{(a,b)}\) must satisfy (1) when \(0 < a < b < 1\), simply because

and the right-hand side tends to \(1 - a - (1-b)) = b-a\) as \(N \rightarrow \infty\).

Using these facts, we show that for any given sequence \(\{x_n\}_{n=1}^\infty\), the class of functions satisfying (1) contains all indicators of the closed intervals \([a,b] \subset [0,1]\) if and only if it contains all continuous, 1-periodic functions \(f\). By linearity, we may assume that \(f\) is real.

Knowing that indicator functions satisfy (1), let \(\mathcal{P}\) be any partition of \([0,1]\) with an odd number of intervals. We can define \(g_{\mathcal{P}}\) and \(h_{\mathcal P}\) as follows: arrange the intervals \(I_1,I_2,\ldots\) of \(\mathcal{P}\) in order from smallest to largest. Modify the intervals so that \(I_1,I_3,\ldots\) are closed and \(I_2,I_4,\ldots\) are open; now every point of \([0,1]\) belongs to exactly one such interval. We can define

and

Both \(f\) and \(g\) are finite linear combinations of functions known to satisfy (1). Because \(f\) is continuous on \([0,1]\), it is uniformly continuous and the functions \(g_{\mathcal P}\) and \(h_{\mathcal P}\) will converge uniformly to \(f\) along any sequence of partitions such that the norm of the partitions tends to zero (i.e., the length of the longest interval).

In the reverse direction, we approximate the indicator function \(\chi_{[a,b]}\) above and below by piecewise linear functions whose integrals are very close to one another. An example is illustrated below assuming that one takes a limit along some sequence of values of \(h\) decreasing to zero.

To show sufficiency of the Weyl criterion, we use the fact that continuous functions can be approximated uniformly by trigonometric polynomials. More precisely, if a sequence is known to be equidistributed, then the Weyl criterion holds simply because \(e^{2 \pi i \ell x}\) is itself a continuous 1-periodic function and its integral on \([0,1]\) happens to be \(0\) when \(\ell \in {\mathbb Z} \setminus \{0\}\).

In the converse direction, if it is known that (1) holds for complex exponentials \(e^{2 \pi i \ell x}\) with \(\ell \in {\mathbb Z} \setminus \{0\}\), then it must hold for all \(\ell \in {\mathbb Z}\) (simply because (1) is true for \(f \equiv 1\) regardless of the sequence). Thus all trigonometric polynomials (i.e., finite linear combinations of complex exponentials) satisfy (1). But all continuous 1-periodic functions are uniformly approximable by trigonometric polynomials, so we can apply the Squeeze Theorem result from above (bounding any real \(f\) between \(q(x) - \epsilon\) and \(q(x) + \epsilon\) for a trigonometric polynomial \(q\) which is uniformly \(\epsilon\)-close to \(f\)) to achieve the desired conclusion.