CONICET | Buscador de Institutos y Recursos Humanos

M. Levin defined a real number $x$ that satisfies that the sequence of the fractional parts of $(2^n x)_{ngeq 1}$ 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 $(2^n x)_{ngeq 1}$ 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 $(2^n x)_{ngeq 1}$ fail to have Poissonian pair correlations.