How slowly can a bounded sequence cluster?

Abstract

We propose a simple measure of how slowly a bounded real sequence clusters. This measure, called \emph{separation}, is the infimum, over all finite segments of the sequence with at least two terms, of the ratio of the least distance between the terms in the segment to the general supremum of such a distance for a segment of that length. An example of a highly separated sequence is given. To create a more separated sequence, we modify van der Corput's construction, replacing the powers of a base by the even-numbered terms of the Fibonacci sequence. The result coincides initially with the sequence built stepwise by maximizing separation for each extra term. We conjecture that these sequences are the same and of maximal separation.