Continued fractions of arithmetic sequences of quadratics

Abstract

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

Introduction

In this paper we address some natural questions that were raised in [2] on the study of the evolution of the continued fraction expansion within a fixed real quadratic field. The reader is referred there for more details and motivation. We review here the necessary definitions for the presentation of our results.

For a real number $x\in [0,1]\setminus \mathbb{Q}$ let $x=[a_1\left(x\right),a_2\left(x\right),\dots ]$ be its uniquely defined continued fraction expansion. For a finite sequence of natural numbers $w=(w_1,\dots ,w_k)$ we define the frequency of appearance of the pattern $w$ in the continued fraction expansion of $x$, by

$$D(x,w)=\underset{N}{lim}\frac{1}{N}\#\left\{1\le n\le N:w=(a_{n+1}\left(x\right),\dots ,a_{n+k}\left(x\right))\right\}.$$

We make the convention that for $x\in \mathbb{R}\setminus \mathbb{Q}$, $D(x,w)≔D(\bar{x},w)$ where $\bar{x}\in [0,1]$ and $\bar{x}=xmod1$. As we explain in the introduction of [2], for Lebesgue almost any $x$, $D(x,w)$ exists and is equal to some explicit integral which depends only on $w$. We denote this almost sure value as $c_w$.

In [2] we fix a quadratic irrational $x$ and study the behaviour of $D(p^{n}x,w)$ as $n\to \infty$ where $p$ is a fixed natural number, or more generally, the behaviour of $D(p_{n}x,w)$ when $p_n\to \infty$ are supported on a fixed finite set of primes $S$ (by supported we mean that if a prime $p$ divides $p_n$ for some $n$ then $p\in S$). The methods of [2] fail when the numbers $p_n$ are not supported of finite set of primes. The aim of this note is merely to rephrase known results in analytic number theory using a refinement of Duke’s Theorem (See Section 2.1) in order to answer question (4) of [2, §2.7] and some related results. Using the results of [4], [16] we have the following unconditional result:

Theorem 1

There exist at most two real quadratic fields such that if a quadratic irrational $x$ does not belong to them, there exist an infinite subset $S\subset \mathbb{P}$ and $\delta >0$ such that for any pattern $w$ there exists a constant $C=C(w,x)$ with

$$p\in S⟹|D(px,w)-c_w|<C{p}^{-\delta }.$$

Using [14] with  [11], [17], we have the following two conditional results:

Theorem 2

Let $x$ be a quadratic irrational. Then, assuming that the Generalized Riemann Hypothesis (GRH) holds, there exist a density one subset $S\subset \mathbb{N}$ and $\delta >0$ such that for any pattern $w$ there exists a constant $C=C(w,x)$ with

$$m\in S⟹|D(mx,w)-c_w|<C{m}^{-\delta }.$$

Similarly, let $P$ denote the set of primes. Assuming that the GRH holds, there exist a density one sequence $S\subset P$ and $\delta >0$ such that for any pattern $w$ there exists a constant $C=C(w,x)$ with

$$p\in S⟹|D(px,w)-c_w|<C{p}^{-\delta }.$$

Remark 3

Using the same methods, Theorem 1, Theorem 2 can be slightly generalized. For example, these theorems remain valid if we can consider $D(w,\frac{x}{m})$ instead of $D(w,mx)$ for $m$ coprime with $\mathrm{disc}\left(\mathbb{Q}\left(x\right)\right)$ (as the associated orders are the same, i.e. ${\mathcal{O}}_{mx}={\mathcal{O}}_{\frac{x}{m}}$, see Section 2.1). Theorem 1, Theorem 2 should be viewed as what we consider to be representative results that can be achieved via these techniques.

As mentioned above the following note is merely a reformulation of known results in analytic number theory. As in [2] the above results follow from equidistribution of geodesic loops on the modular surface. In this context, there is an arithmetically defined sequence of collections ${\mathcal{G}}_d$ of closed geodesics that equidistribute as $d\to \infty$; this statement is now classically known as “Duke’s Theorem”. When one considers the statistics of continued fraction sequence along some arithmetic sequences, one is led to consider subcollections (in fact we will consider one “long” geodesic) of ${\mathcal{G}}_d$. When these subcollections occupy a substantial part of ${\mathcal{G}}_d$, one can hope that they will equidistribute too (c.f. the far-reaching conjecture [6, Conjecture 1.11]). Section 2 is devoted to describe these notions and state a variant of Duke’s Theorem for subcollections. This establishes the bridge between statistics of continued fractions expansion and closed geodesics. Section 3 starts with the simple observations connecting the later to orders of certain matrices. Then we state several results from analytic number theory about these orders (which can be viewed as analogues of Artin’s conjecture on primitive roots) and deduce our results.