Pair correlation for fractional parts of αn2
By D. R. Heath-Brown. Mathematical Proceedings of the Cambridge Philosophical Society, 2009
61 citations recorded by the source corpus.
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)\begin{equation} \left|\alpha-\frac{a}{q}\right|\gg_{\ep} q^{-2-\ep} \end{equation} 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/3fc8ed23754cd2967c9a03c876ac723e8b9ca8fa
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Citations in the corpus
- Modular hyperbolas (2011, background): “However, unfortunately in the range relevant to the pair correlation problem the corresponding results of [109] and [179] are essentially of the same strength.”
- The two-point correlation function of the fractional parts of $\sqrt{n}$ is Poisson (2013, background): “) Other number-theoretic sequences, whose two-point correlations are Poisson, include the values of positive definite quadratic forms subject to certain diophantine conditions [3,12], forms in more variables [15–17], inhomogeneous forms in two [5,8] and more variables [7], and the fractional parts o”