13.12 Rafael Sayous: Pair correlations of fractional powers, two examples

Abstract: Given a family {u_i} in a topological additive group, one may be interested in the statistics obtained by taken all the differences u_i-u_j, in particular for reasons arising from physics (quantum physics for energy level repartitions) or as a pseudo-randomness measurement of the family {u_i}. This approach is known as the study of pair correlations of {u_i}. Fix α a real parameter in ]0,1[. In this talk, I will give and explain two recent examples where the pair correlation exhibit completely different behaviours depending on the scaling: first the sequence (n^α) on the real line (where n takes values in the natural integers), and then the family {n^α} where n takes values in a complex lattice (or grid) and where we use a standard definition for fractional powers of complex points.