Continued fractions of arithmetic sequences of quadratics

Abstract

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite set S⊂P such that the statistics of the period of the continued fraction expansions along the sequence px:p∈S approach the ‘normal’ statistics given by the Gauss–Kuzmin measure. Under the generalized Riemann hypothesis, we prove that there exist full density subsets S⊂P and T⊂N satisfying the same assertion. We give a rate of convergence in all cases. © 2019 Elsevier GmbH