The champernowne constant is not poissonian

Abstract

In this note we study the pair correlation statistics for the sequence of shifts of α, xn = {2n α}, n = 1, 2, 3, . . ., where we choose α as the Champernowne constant in base 2. Throughout this article {·} denotes the fractional part of a real number. It is well known that (xn)n∈N has Poissonian pair correlations for almost all normal numbers α (in the sense of Lebesgue), but we will show that it does not have this property for all normal numbers α, as it fails to be Poissonian for the Champernowne constant.