On Sequences With Exponentially Distributed Gaps

Abstract

It is well known that a sequence which has Poissonian correlations of all orders necessarily has exponentially distributed nearest‐neighbor gaps. It is natural to ask whether this implication also holds in the other direction, that is, whether a sequence with exponential gap distribution must have Poissonian correlations, and by an already known fact, must be equidistributed. We show that this assertion is generally false, by constructing a sequence that has exponential gap distribution but fails to be equidistributed (and as a consequence, also fails to have Poissonian correlations of any order and scale).