Pair correlation for fractional parts of αn2

It was proved by Weyl [8] in 1916 that the sequence of values of αn2 is uniformly distributed modulo 1, for any fixed real irrational α. Indeed this result covered sequences αnd for any fixed positive integer exponent d. However Weyl's work leaves open a number of questions concerning the finer distribution of these sequences. It has been conjectured by Rudnick, Sarnak and Zaharescu [6] that the fractional parts of αn2 will have a Poisson distribution provided firstly that α is “Diophantine”, and secondly that if a/q is any convergent to α then the square-free part of q is q1+o(1). Here one says that α is Diophantine if one has (1.1) for every rational number a/q and any fixed ϵ > 0. In particular every real irrational algebraic number is Diophantine. One would predict that there are Diophantine numbers α for which the sequence of convergents pn/qn contains infinitely many squares amongst the qn. If true, this would show that the second condition is independent of the first. Indeed one would expect to find such α with bounded partial quotients.

Uniformly Diophantine numbers in a fixed real quadratic field

The field$\mathbb {Q}(\sqrt {5})$contains the infinite sequence of uniformly bounded continued fractions$[\overline {1,4,2,3}], [\overline {1,1,4,2,1,3}], [\overline {1,1,1,4,2,1,1,3}], \ldots ,$and similar patterns can be found in$\mathbb {Q}(\sqrt {d})$for anyd>0. This paper studies the broader structure underlying these patterns, and develops related results and conjectures for closed geodesics on arithmetic manifolds, packing constants of ideals, class numbers and heights.