On Sequences With Exponentially Distributed Gaps

It is well known that a sequence (Formula presented.) 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).