Integrals and Seriesby Shobhit Bhatnagar

Number Theory

Weyl's Equidistribution Theorem

Shobhit Bhatnagar

Weyl's criterion lets us test uniform distribution by checking exponential sums. We will prove the equivalence between this test, interval counts, and averages of Riemann-integrable functions. A sequence of real numbers \(x_1, x_2, \cdots\) is said to be equidistributed (mod 1) if for every sub-interval \((a,b)\subset [0,1]\), we have

\[ \lim_{N\to \infty}\frac{|\{1\leq n\leq N:\; \langle x_n \rangle\in (a,b)\}|}{N} = b-a \]

where \(\langle x \rangle\) denotes the fractional part of \(x\). Weyl's equidistribution criterion states that the following statements are equivalent:

  1. \(x_1, x_2, \cdots\) are equidistributed (mod 1).

  2. For each non-zero integer \(k\), we have

    \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N e^{2\pi i k x_n}=0 \]

  3. For each Riemann integrable function \(f:[0,1]\to\mathbb{C}\), we have

    \[ \lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^N f(\langle x_n \rangle) = \int_0^1 f(x) dx \]

Proof: (1) ⇒ (3)

Let \(I=[a,b)\subseteq [0,1]\) and note that

\[ \frac{|\{1 \leq n \leq N: \langle x_n \rangle\in [a,b) \}|}{N}=\frac{1}{N}\sum_{n=1}^N \chi_{[a,b)}(\langle x_n \rangle) \]

where \(\chi_{[a,b)}(x)\) equals 1 if \(x\in [a,b)\) and 0 otherwise. This shows that (3) holds for the case when \(f\) is a characteristic function. Now, let \(\lambda_1, \lambda_2\in \mathbb{R}\) and \(f_1\), \(f_2\) be functions for which (3) holds. Then,

\[ \begin{aligned}\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^N (\lambda_1 f_1 + \lambda_2 f_2)(\langle x_n \rangle) &= \lim_{N\to \infty} \frac{\lambda_1}{N}\sum_{n=1}^N f_1(\langle x_n\rangle) + \lim_{N\to \infty}\frac{\lambda_2}{N}\sum_{n=1}^Nf_2(\langle x_n\rangle) \\ &= \lambda_1\int_0^1 f_1(x) dx + \lambda_2 \int_0^1 f_2(x) dx \\ &= \int_0^1 (\lambda_1 f_1 + \lambda_2 f_2)(x) dx\end{aligned} \]

Thus, (3) holds for all linear combinations of characteristic functions of subintervals of \([0,1]\).

Now, let \(f:[0,1]\to \mathbb{R}\) be an integrable function, and let \(\epsilon >0\). Choose step functions \(f_1\) and \(f_2\) such that:

  • \(f_1\leq f\leq f_2\) pointwise
  • \(\int_0^1 (f_2(x)-f_1(x))dx < \frac{\epsilon}{2}\)
  • There exists \(N_0\) such that \(\left|\int_0^1 f_1(x)dx - \frac{1}{N}\sum_{n=1}^N f_1(\langle x_n\rangle) \right| < \frac{\epsilon}{2}\) and \(\left|\int_0^1 f_2(x)dx - \frac{1}{N}\sum_{n=1}^N f_2(\langle x_n\rangle) \right| < \frac{\epsilon}{2}\) for all \(N\geq N_0\)

It follows that for \(N\geq N_0\),

\[ \begin{aligned} \int_0^1 f(x) dx - \frac{1}{N}\sum_{n=1}^N f(\langle x_n\rangle) &\leq \int_0^1 f(x) dx - \frac{1}{N}\sum_{n=1}^N f_1(\langle x_n\rangle) \\ &< \int_0^1 f(x) dx -\int_0^1 f_1(x) dx +\frac{\epsilon}{2} \\ &< \int_0^1 (f_2(x)-f_1(x)) dx + \frac{\epsilon}{2} \\ &< \epsilon \end{aligned} \]

In a similar way, we can prove that

\[ \int_0^1 f(x) dx - \frac{1}{N}\sum_{n=1}^N f(\langle x_n\rangle) > -\epsilon \quad \forall\; N\geq N_0 \]

Therefore, we have

\[ \left|\int_0^1 f(x) dx - \frac{1}{N}\sum_{n=1}^N f(\langle x_n\rangle) \right| < \epsilon \quad \forall\; N\geq N_0 \]

To see that (3) holds when \(f\) is complex valued, we need only consider the real and imaginary parts separately.

(2) ⇒ (3)

First let \(f\) be continuous and \(1\)-periodic. Trigonometric polynomials uniformly approximate such functions on the circle. For any \(\epsilon>0\), choose

\[ p(x)=\sum_{|k|\leq m}c_k e^{2\pi i kx},\qquad \|f-p\|_\infty<\epsilon. \]

By (2), the average of \(p(x_n)\) tends to its constant coefficient \(c_0=\int_0^1p(x)\,dx\). Thus

\[ \limsup_{N\to\infty}\left|\frac1N\sum_{n=1}^N f(\langle x_n\rangle)-\int_0^1 f(x)\,dx\right|\leq2\epsilon. \]

Letting \(\epsilon\to0\) proves the assertion for continuous periodic \(f\).

To pass to interval indicators, fix an arc \(I\) of the circle. For any \(\epsilon>0\), choose continuous periodic functions \(g_-\) and \(g_+\), with short linear ramps near the endpoints, such that

\[ g_-\leq\chi_I\leq g_+,\qquad \int_0^1(g_+-g_-)\,dx<\epsilon. \]

Their integrals bracket the length of \(I\). Applying the result just proved to \(g_-\) and \(g_+\) squeezes the limiting interval frequency to that length. The same construction with a small neighborhood of a single point shows that endpoint visits have density zero, so the choice of open, closed, or half-open endpoints does not affect the result. This proves (1). The argument in (1) ⇒ (3) now extends the conclusion to all Riemann-integrable functions.

Periodicity is essential in the first approximation step: a continuous function on \([0,1]\) whose endpoint values differ cannot be uniformly approximated there by trigonometric polynomials.

The implications (3) ⇒ (1) and (3) ⇒ (2) are obvious.

References

  • Hannigan-Daley, Brad. Equidistribution and Weyl’s criterion. Retrieved from http://individual.utoronto.ca/hannigandaley/equidistribution.pdf. Accessed 5 Feb. 2020.
  • Stein, Elias M. and Shakarchi, Rami. Fourier Analysis: An Introduction. Princeton University Press, 2003
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