The distribution of spacings between the fractional parts of n² α

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.