Low discrepancy sequences failing Poissonian pair correlations

Abstract

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