# Continued Fractions of Arithmetic Sequences of Quadratics

## TL;DR

This paper investigates the continued fraction expansions of quadratic irrationals scaled by primes, demonstrating that their statistical properties align with the Gauss-Kuzmin distribution along certain prime subsets, under specific hypotheses.

## Contribution

It establishes the existence of prime subsets where the continued fraction period statistics of scaled quadratic irrationals follow the Gauss-Kuzmin measure, including under the generalized Riemann hypothesis.

## Key findings

- Existence of infinite prime subsets with normal continued fraction statistics.
- Under GRH, full density subsets exhibit the same statistical behavior.
- Provides explicit convergence rates for the distribution of continued fraction periods.

## Abstract

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in 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 of P and T of N satisfying the same assertion. We give a rate of convergence in all cases.