Franklin's Notes

Weyl criterion

The Weyl criterion is the following equivalent condition for a sequence $(x_n)$ of real numbers to be uniformly distributed modulo 1 :

Theorem 1. The sequence $(x_n)$ is uniformly distributed modulo $1$ if and only if for all integers $h\ne 0$.

Proof. Necessity of this condition follows directly from Proposition 5 here and the fact that the functions $x\mapsto e^{2\pi i h x}$ are continuous functions of $x$ with period $1$, so long as $h\in\mathbb Z\backslash{0}$. Further, by the Weierstrass approximation theorem, for any $\epsilon>0$, for any continuous function $f:\mathbb R\to\mathbb R$ with period $1$, there exists a trigonometric polynomial $g$ such that $|f(x)-g(x)|\leq\epsilon/3$ for all $x\in [0,1]$. Now, we have by the triangle inequality that where We have that $E_1,E_3\leq \epsilon/3$ by the definition of $g$, and if we assume that the aforementioned limits of sums of imaginary exponentials hold true, then we have that $N$ may be chosen sufficiently large such that $E_2\leq \epsilon/3$. This enforces showing that this difference can be made arbitrarily small by choosing $N$ sufficiently large, and so we have that for the arbitrarily chosen continuous function $f:\mathbb R\to\mathbb R$ with period $1$. By Proposition 5 , it follows that $(x_n)$ is uniformly distributed modulo $1$, so the criterion of Weyl is also sufficient. $\blacksquare$






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