  M. Levin defined a real number $x$ that satisfies that the sequence of the
fractional parts of $(2^n x)_{n\geq 1}$ are such that the first $N$ terms have
discrepancy $O((\log N)^2/ N)$, which is the smallest discrepancy known for
this kind of parametric sequences. In this work we show that the fractional
parts of the sequence $(2^n x)_{n\geq 1}$ fail to have Poissonian pair
correlations. Moreover, we show that all the real numbers $x$ that are variants
of Levin's number using Pascal triangle matrices are such that the fractional
parts of the sequence $(2^n x)_{n\geq 1}$ fail to have Poissonian pair
correlations.
