Exercise 58, § 1

For every nonzero real number $\lambda$, let $\mu_{\lambda}$ denote the discrete probability measure on $\mathbf{R}$ such that $\mu_{\lambda}(\lambda ) =\mu_{\lambda}(-\lambda ) =\frac{1}{2}$.

Fix $\lambda \in ]0,1[$.

a) Put $\nu_0=\mu_1$ and $\nu_{n+1}=\mu_{\lambda^{n+1}}*\nu_n$ for every integer $n\geqslant 0$. The sequence $(\nu_n)_{n\in\mathbf{N}}$ converges to a probability measure $\nu_{\lambda}$ on $\mathbf{R}$. The characteristic function of $\nu_{\lambda}$ is the function on $\mathbf{R}$ defined by

$\varphi_{\lambda}(x) =\prod_{n=0}^{+\infty}$ cos($\lambda^nx$).

b) The measure $\nu_{\lambda}$ is the unique positive measure $\nu$ of total mass 1 on $\mathbf{R}$ such that

where $\ell_1$ (resp. $\ell_2$) is the mapping $x\mapsto \lambda x-1$ (resp. $x\mapsto \lambda x+ 1$) of $\mathbf{R}$ into $\mathbf{R}$.

c) If the measure $\nu_{\lambda}$ is not a measure of base Lebesgue measure, then $\nu_{\lambda}$ is singular with respect to Lebesgue measure. (Consider the Lebesgue decomposition of $\nu_{\lambda}$ with respect to Lebesgue measure, cf. INT, V, §5, n$^o7$, th. 3.)

d) If $0< \lambda <\frac{1}{2}$, then the measure $\nu_{\lambda}$ is singular with respect to Lebesgue measure. e) If $\lambda =\frac{1}{2}$, then the measure $\nu_{\lambda}$ is equal to the uniform measure on $[-2,2]$. If $\lambda >1/2$, the support of $\nu_{\lambda}$ is the interval $[-(1-\lambda )^{-1},(1-\lambda )^{-1}]$.

f) A Pisot number is a real algebraic integer $x$ such that all the conjugates $y\in \mathbf{C}$ of $x$, other than $x$, satisfy $|y|<1$. If $\lambda^{-1}$ is a Pisot number, then the function $\varphi_{\lambda}$ does not tend to 0 at infinity. In particular, the measure $\nu_{\lambda}$ is singular with respect to Lebesgue measure.

g) There exists $\lambda >1/2$ such that $\nu_{\lambda}$ is singular with respect to Lebesgue measure.

Henceforth assume that $\varphi_{\lambda}$ does not tend to 0 at infinity, and put $\theta = 1/\lambda$. For every real number $x$, let $\langle x\rangle$ denote the unique real number in $[0,1[$ such that $x- \langle x\rangle \in \mathbf{Z}$.

h) There exists a real number $u\not= 0$ such that the series with general term sin($u\theta^n$)$^2$ converges; the series with general term $\langle u\theta^n\rangle$ converges.

i) For every $n\in \mathbf{N}$, let $c_n$ be an integer such that $|u\theta^n-c_n|\leqslant \frac{1}{2}$. The formal series $g(z) =\sum c_nz^n$ is a rational fraction. (Use the criterion of A, IV, p. 85, exercise 1, and Hadamard's inequality, EVT, V, p. 37, cor. 3, to show that the Hankel determinants which arise tend to 0; conclude by observing that these determinants are integers.)

j) There exist polynomials with integer coefficients $f_1$ and $f_2$ such that $g=f_1/f_2$ and $f_2(0) = 1$.

k) The number $\theta$ is a Pisot number ("Salem's theorem"). (Observe that $f_2(\lambda ) = 0$ and that the other roots of $f_2$ have modulus $>1$.)