Abstract:

Questions about how "random" or "deterministic" a sequence is appearing in many areas of mathematics.  For real-valued sequences, this can be formulated in terms of how well distributed points are in the real line, or in an interval.  More detailed information can be found if we look at the gaps between pairs of points in finite segments of such sequences.  If these are sufficiently random then the mean gap data converges in a natural way, and the sequence has "Poisson pair correlations" (PPC) introduced by Rudnick and Sarnak.  There is a rich literature on such gaps of pairs in sequences studied in number theory: for example, the sequence $(\alpha n) _n$ for $\alpha$ irrational is equidistributed modulo 1 for any irrational $\alpha$, but does not have PPC.