Remark on a result of Bourgain on poissonian pair correlation

Abstract: We show for a class of sequences $(a_n){n\geq 1}$ of distinct positive integers, that for no $\alpha$ the sequence $(\left{a_n \alpha \right}){n \geq 1}$ does have Poissonian pair correlation. This class contains for example all strictly increasing integer sequences with positive upper density. This result motivates us to state a certain conjecture on Poissonian pair correlation which would be a significantly stronger version of a result of Jean Bourgain.