Exercise 35, § 1

a) For every integer $N\geqslant 1$, every family $(a_n)_{1\leqslant n\leqslant N}$ of complex numbers and every integer $H\geqslant 1$, one has

$n=1|h|<H1\leqslant n\leqslant N$

(“van der Corput inequality”). (Set $a_n= 0$ if $n$ is not between 1 and N; write $\sum_na_n=\sum_na_{n+h}$ for $0\leqslant h <H$, sum over $h$, then apply the Cauchy-Schwarz inequality.)

Let $\pi :\mathbf{R}\rightarrow \mathbf{R}/\mathbf{Z}$ be the canonical projection. We say that a sequence $(x_n)$ of real numbers is equidistributed modulo 1 if the sequence $(\nu_n)$ of measures on $\mathbf{R}/\mathbf{Z}$ defined by

$n$

1

$k=1$

converges vaguely towards the Haar measure of total mass 1 on $\mathbf{R}/\mathbf{Z}$. b) If, for every integer $h\geqslant 1$, the sequence $(x_{n+h}-x_n)_n$ is equidistributed modulo 1, then $(x_n)$ is equidistributed modulo 1. (Use the preceding exercise and the van der Corput inequality.)

c) Let $d\geqslant 1$ be an integer and let $P =\alpha_dX^d+\cdots +\alpha_0\in \mathbf{R}[X]$ be a polynomial such that $\alpha_d\notin\mathbf{Q}$. Then the sequence $(P(n))$ is equidistributed modulo 1. (Reason by induction on $d\geqslant 1$.)