On the number of gaps of sequences with Poissonian pair correlations

Abstract A sequence ( x n ) on the unit interval is said to have Poissonian pair correlation if # { 1 ≤ i ≠ j ≤ N : ‖ x i − x j ‖ ≤ s / N } = 2 s N ( 1 + o ( 1 ) ) for all reals s > 0 , as N → ∞ . It 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 any index subsequence ( n t ) . First, we improve this by showing that, if ( x n ) has Poissonian pair correlations, then 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 such that g ( n ) ≤ f ( n ) for all sufficiently large n. This answers negatively a question posed by G. Larcher.