  We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian
pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \#
\lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \rbrace = 2s
\end{equation*} for every $s \geq 0$. In this note we study the pair
correlation statistics for the sequence of shifts of $\alpha$, $x_n = \lbrace
2^n \alpha \rbrace, \ n=1, 2, 3, \ldots$, where we choose $\alpha$ as the
Champernowne constant in base $2$. Throughout this article $\lbrace \cdot
\rbrace$ denotes the fractional part of a real number. It is well known that
$(x_n)_{n \in \mathbb{N}}$ has Poissonian pair correlations for almost all
normal numbers $\alpha$ (in the sense of Lebesgue), but we will show that it
does not have this property for all normal numbers $\alpha$, as it fails to be
Poissonian for the Champernowne constant.
