  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.
