On the number of gaps of sequences with Poissonian Pair Correlations

Abstract

A sequence (x n ) on the torus is said to have Poissonian pair correlations ifIt is known that, if (x n ) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x 1 , . . . , x n } cannot be bounded along every index subsequence (n t ). First, we improve this by showing that the maximum among the multiplicities of the neighboring gap lengths of {x 1 , . . . , x n } is o(n), as n → ∞. Furthermore, we show that, for every function f : N + → N + with lim n f (n) = ∞, there exists a sequence (x n ) with Poissonian pair correlations and such that g(n) ≤ f (n) for all sufficiently large n. This answers negatively a question posed by G. Larcher.