The distribution of spacings of real?valued lacunary sequences modulo one

Let ( a n ) = 1 ? $(a_{n})_{n=1}^{\infty }$ be lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all ? ? R $\alpha \in \mathbb {R}$ , the pair correlation $(\alpha a_{n})_{n=1}^{\infty mod is Poissonian. We show higher correlations hence nearest-neighbour spacing distribution are Poissonian as well, thereby extending result Zaharescu to real-valued sequences.