Quadratic Lagrange spectrum: I

TL;DR

This paper establishes the existence of Hall's ray for quadratic Lagrange spectra of all real quadratic numbers and computes Hurwitz constants for many such spectra, advancing understanding of Diophantine approximation.

Contribution

It proves the existence of Hall's ray for quadratic Lagrange spectra and calculates Hurwitz constants for a broad class of real quadratic numbers, a novel extension in the field.

Findings

Hall's ray exists for all real quadratic numbers' spectra

Hurwitz constants are computed for many quadratic Lagrange spectra

Provides new insights into Diophantine approximation of quadratic irrationals

Abstract

In this paper we prove the existence of Hall's ray for the quadratic Lagrange spectrums of all real quadratic numbers. For a large class of real quadratic numbers, we compute the Hurwitz constants of their quadratic Lagrange spectrums

Equations

$$|\alpha -\frac{p}{q}|<\frac{1}{q^2}.$$

$$L(\alpha ):=\underset{q\to \infty }{\mathrm{liminf}}q|q\alpha -p|.$$

$$c_{\alpha }\xi :=\underset{\genfrac{}{}{0ex}{}{\beta \in {\Theta }_{\alpha }}{|\beta -{\beta }^{\sigma }|\to 0}}{\mathrm{liminf}}2\frac{|\xi -\beta |}{|\beta -{\beta }^{\sigma }|}$$

$$\underset{v\to \infty }{\lim }{K}_{{\beta }_{u,v}}=\frac{3}{\sqrt{5}}-1-\frac{2}{u\sqrt{5}}$$

$$K_{{\beta }_{u,v}}=\frac{2({\tau }_1-{\phi }^2)(1+\frac{{\tau }_2}{{\phi }^2})}{(\phi +\frac{1}{\phi })({\tau }_1{\tau }_2+1)},$$

$$K_{\alpha }=\max \{c_{\alpha }\phi ,\underset{m\to \infty }{\lim }{c}_{\alpha }{\phi }_m\},$$

$$[a_0,a_1,a_2,\cdots ,a_n]$$

$$a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots +\frac{1}{a_n}}},$$

$$[a_0,a_1,a_2,\cdots ,a_n,\cdots ]$$

$$a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots +\frac{1}{a_n+\cdots }}},$$

$$[a_0,a_1,a_2,\cdots ,a_{k-1},\overline{a_k,\cdots ,a_{k+m-1}}],$$

$$a=[a_0;a_1,a_2,\cdots ,a_n,\cdots ]$$

$$p_{-2}=0,p_{-1}=1,p_n=a_n{p}_{n-1}+p_{n-2}(n\ge 0),$$

$$q_{-2}=1,q_{-1}=0,q_n=a_n{q}_{n-1}+q_{n-2}(n\ge 0).$$

$$q_{m+n}\ge 2^{\frac{m-1}{2}}{q}_n,$$

$$[a_0;a_1,a_2,\cdots ,a_n]=\frac{p_n}{q_n},$$

$$|a-\frac{p_n}{q_n}|<\frac{1}{q_n{q}_{n+1}}.$$

$$\alpha =[a_0,a_1,a_2,\cdots ],$$

$$\beta =[b_0,b_1,b_2,\cdots ]$$

$$\frac{1}{72q_n{q}_{n+2}}\le \frac{1}{72q_n^2{a}_{n+1}{a}_{n+2}}\le |\alpha -\beta |.$$

$$\frac{1}{72(C+1{)}^{2n+2}}\le |\alpha -\beta |.$$

$$\alpha =[a_0,a_1,a_2,\cdots ],$$

$$\beta =[b_0,b_1,b_2,\cdots ],$$

$$\gamma =[c_0,c_1,c_2,\cdots ]$$

$$|\alpha -\beta |\le \frac{72}{2^{m-n-3}}|\alpha -\gamma |.$$

$$|\alpha -\beta |\le 4\cdot 72D^4|\alpha -\gamma |.$$

$$|\alpha -\beta |\le \max (\frac{1}{q_m{q}_{m-1}},\frac{1}{q_m^{\prime }{q}_{m-1}^{\prime }})\le \frac{1}{2^{m-n-3}{q}_n{q}_{n+2}}\le \frac{72}{2^{m-n-3}}|\alpha -\gamma |.$$

$$|\alpha -\beta |\le \max (\frac{1}{q_m{q}_{m-1}},\frac{1}{q_m^{\prime }{q}_{m-1}^{\prime }})\le \frac{1}{q_{m-1}^2}\le \frac{4D^2}{q_n{q}_{n+2}}\le 4\cdot 72D^2|\alpha -\gamma |.$$

$${\alpha }^{\prime }=[a_m,a_{m+1},a_{m+2},\cdots ],$$

$${\beta }^{\prime }=[b_m,b_{m+1},b_{m+2},\cdots ],$$

Full text

*College of Mathematics and Computer Science, Fujian Normal University,

*  *Fuzhou, 350108, China;

*  Email: linxianzu@126.com

Keywords: continued fractions, Lagrange spectrums, Hall’s ray, Hurwitz constant.

Mathematics Subject Classification 2010: 11J06, 11J70.

1. Introduction

By the classical Dirichlet theorem, for any irrational number $\alpha$, there exist infinitely many integers $p,q$ such that

[TABLE]

If we want to strengthen this ineqality by replacing $1$ with a smaller constant, we need to consider

[TABLE]

The classical Lagrange spectrum is defined to be the set L of values $L(\alpha)$ where $\alpha$ runs over all irrational numbers. In 1879, Markoff [6, 7] proved that L begins with a discrete sequence: $\tfrac{1}{\sqrt{5}},$ $\tfrac{1}{\sqrt{8}}$, $\tfrac{\sqrt{221}}{5},\cdots$ which converges to $\tfrac{1}{3}$, where $\tfrac{1}{\sqrt{5}}$ is the Hurwitz constant. In 1947, Hall [4] showed that L contains a nontrivial interval $[0,\epsilon](Hall^{\prime}s\ ray)$. In 1975, Cusick [2] proved that L is a closed subset of $\mathbb{R}$. We refer the readers to [3] and the references therein for more properties of Lagrange spectrum.

In [9, 10], as a corollary of their geometric generalizations of the Lagrange Spectra in negative curvature, Parkkonen and Paulin defined the quadratic Lagrange spectrum as follows

Throughout this paper, let σ be the Galois conjugate of a quadratic number $x$. Let $\alpha$ be a fixed real quadratic number, and let $\Theta_{\alpha}=PSL(2,\mathbb{Z})\{\alpha,\alpha^{\sigma}\}$ be the orbit of $\alpha$ and $\alpha^{\sigma}$ for the action of $PSL(2,\mathbb{Z})$. The quadratic Lagrange spectrum of $\alpha$ is defined to be the set $Sp_{\alpha}$ of values

[TABLE]

where $\xi$ runs over all real number not in $\mathbb{Q}\cup\Theta_{\alpha}$.

Parkkonen and Paulin [9, 10] showed that $Sp_{\alpha}$ is a closed subset of $\mathbb{R}$. But their geometric method does not imply the existence of Hall’s ray for $Sp_{\alpha}$. The first main result of this paper is existence of Hall’s ray for $Sp_{\alpha}$:

Theorem 1.1**.**

For any real quadratic number $\alpha$, there exists a positive number $\psi=\psi(\alpha)$ (see Section 3 for the definition of $\psi(\alpha)$) such that $[0,\psi]\subset Sp_{\alpha}$.

Define the Hurwitz constant $K_{\alpha}$ of $\alpha$ to be the maximum of $Sp_{\alpha}$, Parkkonen and Paulin [9, 10] showed that $K_{\alpha}\leq(1+\sqrt{2})\sqrt{3}$ for each real quadratic number $\alpha$. In [1], Bugeaud pointed out that the theory of continued fractions is well suited for the investigation of quadratic Lagrange spectrum. Among other results, he showed that $K_{\alpha}\leq\tfrac{1}{2}$ and $K_{\varphi}=\tfrac{3}{\sqrt{5}}-1$, where $\varphi$ is the Golden Ratio $(1+\sqrt{5})/2$. Bugeaud conjectured that $\tfrac{3}{\sqrt{5}}-1$ is a common upper bound for all the $K_{\alpha}$. Pejković[8] proved this conjecture using the theory of continued fractions. He also showed that

[TABLE]

where $\beta_{u,v}=[\overline{u,v}]$ and $v\geq u>1/0.0008$.

In this paper, we determine the Hurwitz constant $K_{\beta_{u,v}}$ for $v\geq u\geq 9$.

Theorem 1.2**.**

For $v\geq u\geq 9$, the Hurwitz constant

[TABLE]

where $\tau_{1}=[\overline{u,v}]$ and $\tau_{2}=[\overline{v,u}]$.

Remark 1.3**.**

In [1], Bugeaud conjectured the following formula for $K_{\alpha}$

[TABLE]

where $\varphi_{m}=[\overline{m}]$. We calculate both $c_{\alpha}\varphi$ and $\lim_{m\rightarrow\infty}c_{\alpha}\varphi_{m}$ for $\alpha=\beta_{u,v}$, and prove that the maximal of them is indeed $K_{\beta_{u,v}}$ when $v\geq u\geq 9$. In fact a more delicate method shows that Theorem 1.2 also holds when $v\geq u\geq 4$.

Remark 1.4**.**

Our proof of Theorem 1.2 also implies that $K_{\beta_{u,v}}$ is an isolated point of $Sp_{\beta_{u,v}}$, when $v\geq u\geq 9$. It is interesting to see whether $Sp_{\beta_{u,v}}$ begins with a discrete sequence, as in the case of classical Lagrange spectrum.

This paper is structured as follows: In Section 2, we give preliminaries that will be used throughout this paper. In Section 3, we give proof of Theorem 1.1. In Section 4, we determine the Hurwitz constant $K_{\beta_{u,v}}$ for $v\geq u\geq 9$. The author thanks professor Pejković for his careful reading of this paper, and numerous suggestions and corrections for improvement.

2. preliminaries

In this paper, we write

[TABLE]

for the finite continued fraction expansion

[TABLE]

and write

[TABLE]

for the infinite continued fraction expansion

[TABLE]

where $a_{1},a_{2},\cdots,$ are positive integers and $a_{0}$ is an integer. An eventually periodic continued fraction is written as

[TABLE]

where $a_{0},a_{1},a_{2},\cdots,a_{k-1}$ is the preperiod and $a_{k},\cdots,a_{k+m-1}$ is the shortest period.

The sequence of convergents of

[TABLE]

is defined by

[TABLE]

[TABLE]

A direct proof by induction shows that

[TABLE]

for $m,n\geq 1.$

Lemma 2.1**.**

(cf.[5])

[TABLE]

[TABLE]

Lemma 2.2**.**

Let

[TABLE]

and

[TABLE]

be the continued fraction expansions of two real numbers, and let $\{\frac{p_{n}}{q_{n}}\}_{n\geq 0}$ the sequence of convergents of $\alpha$. Let $n$ be a nonnegative integer such that $a_{i}=b_{i}$ for $i=1,\cdots,n-1$, and $a_{n}\neq b_{n}$. Then we have

[TABLE]

Moreover, if there exists a positive integer $C$ such that $|a_{i}|\leq C$, then we have

[TABLE]

Proof.

The first inequality follows directly from [1, Lemma 2.2] and its proof. The second follows from the first since $q_{n}\leq(C+1)^{n}$. ∎

Lemma 2.3**.**

Let

[TABLE]

[TABLE]

and

[TABLE]

be the continued fraction expansions of three real numbers. Let $m$ be a positive integer such that $a_{i}=b_{i}$ for $i=1,\cdots,m-1$, and $a_{m}\neq b_{m}$, and let $n$ be a positive integer such that $a_{i}=c_{i}$ for $i=1,\cdots,n-1$, and $a_{n}\neq c_{n}$. Then if $m>n+2$, we have

[TABLE]

If $0\leq m-n\leq 2$, $n\geq 2$ and $a_{m},a_{m+1},a_{m+2}\leq D$, we have

[TABLE]

Proof.

Let $\{\frac{p_{i}}{q_{i}}\}_{i\geq 0}$ be the sequence of convergents of $\alpha$ and let $\{\frac{p^{\prime}_{i}}{q^{\prime}_{i}}\}_{i\geq 0}$ be the sequence of convergents of $\beta$. Then by (2) and Lemmas 2.1 and 2.2, if $m>n+2$,

[TABLE]

Now we assume that $a_{m},a_{m+1},a_{m+2}\leq D$. If $m-n=1,2$, we have

[TABLE]

If $m=n$, set

[TABLE]

[TABLE]

and

[TABLE]

Then we have

[TABLE]

[TABLE]

and

[TABLE]

As the function $\frac{p_{m-1}x+p_{m-2}}{q_{m-1}x+q_{m-2}}$ is monotone, we have

[TABLE]

On the other hand, by Lemma 2.2,

[TABLE]

Hence the second assertion of the lemma follows. ∎

The next lemma follows directly from an elementary calculation.

Lemma 2.4**.**

Let

[TABLE]

and

[TABLE]

be two doubly infinite sequences of positive integer, satisfying $a_{i}=b_{i}$ for $m\leq i\leq n$. Set

[TABLE]

Then we have

[TABLE]

We recall that the continued fraction expansion of a real quadratic number $\alpha$ is eventually periodic. It is also well-known that if there exists an $A\in PSL(2,\mathbb{Z})$ such that $\beta=A\alpha$ , then the continued fraction expansions of $\alpha$ and $\beta$ have the same tail. Thus throughout this paper, we assume without loss of generality that $\alpha=[\overline{b_{1},\cdots,b_{s}}]$ where $b_{1},\cdots,b_{s}$ and $s$ are positive integers. We extend the notation $b_{i}$ to all $i\in\mathbb{Z}$ by requiring $b_{i}=b_{j}$ whenever $i\equiv j\ (mod\ s)$. It is well-known that the Galois conjugate of $\alpha$ is

[TABLE]

Let $B=2+max(b_{1},b_{2},\cdots,b_{s})$.

The following lemma is from [1, Corollary 2.3].

Lemma 2.5**.**

Let

[TABLE]

be an irrational real number. Let

[TABLE]

where $t\geq 1$, $a_{r+1}\neq a^{\prime}_{1}$ and $a^{\prime}_{t}\neq b_{i}$. Then we have

[TABLE]

Set

[TABLE]

[TABLE]

and set

[TABLE]

For each $\phi\in\Phi$, set

[TABLE]

By Lemma 2.5, when $c_{\alpha}\xi<\tfrac{1}{48B^{3}(B+1)^{2}},$ we have

[TABLE]

Now we repeat the calculations in (cf.[1, 8]) to get an explicit expression for $2\tfrac{|\xi-\xi_{r,\phi}|}{|\xi_{r,\phi}-\xi_{r,\phi}^{\sigma}|}.$

Let $\{\frac{p^{\prime}_{n}}{q^{\prime}_{n}}\}_{n\geq 0}$ be the sequence of convergents of $[a_{0},a_{1},a_{2},\cdots]$. Then by an elementary property of continued fractions (cf.[5, p.133]) we have

[TABLE]

Hence

[TABLE]

Set

[TABLE]

and

[TABLE]

Then

[TABLE]

Hence we have

[TABLE]

3. Hall’s ray for quadratic Lagrange spectrum

In this section, we prove the existence of Hall’s ray for quadratic Lagrange spectrum. Let $n$ be the minimal positive integer satisfying

[TABLE]

It follows from (5) that $n\geq 3$. Set $\psi=\frac{1}{72\cdot 72B^{s(6n+6)+4}}$ and let $\epsilon\in(0,\psi]$.

Lemma 3.1**.**

[TABLE]

defines an increasing function $f_{\epsilon}(x)=y$ which maps $(0,+\infty)$ injectively into $(\alpha-1/2,\alpha+1/2)$. On the interval $[B,B+2]$, we have $M/2\leq f^{\prime}_{\epsilon}(x)\leq 2M$ where

[TABLE]

Proof.

Solving the equation, we get

[TABLE]

A direct calculation shows that

[TABLE]

Hence the lemma follows. ∎

Lemma 3.2**.**

For each $\epsilon\in(0,\psi]$ There exist a positive integer $k>3n$ and two positive infinite continued fractions

[TABLE]

and

[TABLE]

such that

1. i

equation (6) is valid, i.e., $f_{\epsilon}(x)=y$; 2. ii

$(f_{1},f_{2},\cdots,f_{s})\neq(b_{1},\cdots,b_{s})$; 3. iii

any subblock of $[\overline{b_{1}\cdots b_{s}}]$ of length $>ns$ does not occurs in $e_{1}e_{2}\cdots$ and $f_{m+1}f_{m+2}\cdots$.

Proof.

Let $\Gamma$ be the uncountable set of infinte words of elements from $\{B,B+1\}$. Choose a $\omega_{0}=u^{(0)}_{1}u^{(0)}_{2}\cdots\in\Gamma$ and set $\mu_{0}=[u^{(0)}_{1},u^{(0)}_{2},\cdots]$ such that $\nu_{0}=f_{\epsilon}(\mu_{0})$ is irrational, i.e., $\nu_{0}$ has an infinite continued fraction expansion

[TABLE]

where $k\geq 0$ indicates the number of times the block $b_{1},\cdots,b_{s}$ is repeated. and

[TABLE]

Let $\{\frac{p_{n}}{q_{n}}\}_{n\geq 1}$ be the sequence of convergents of $[\overline{b_{1},\cdots,b_{s}}]$. By Lemma 2.2 and (6) we have

[TABLE]

hence $k>3n$.

Claim 3.3**.**

For any $\omega_{1}=v^{(1)}_{1}v^{(1)}_{2}\cdots\in\Gamma$,

[TABLE]

lies in the interval $f_{\epsilon}([u^{(0)}_{1},u^{(0)}_{1}+1])$.

Proof.

The distance of $[u^{(0)}_{1},u^{(0)}_{2},\cdots]$ from the boundary of $[u^{(0)}_{1},u^{(0)}_{1}+1]$ is at least $\frac{1}{B+2}$. If the claim is invalid, then by Lemma 3.1 we have

[TABLE]

On the other hand, by Lemma 2.3, we have

[TABLE]

Hence

[TABLE]

This contradicts (5). ∎

Now, pick $\nu_{1}=[\underbrace{b_{1},\cdots,b_{s}}_{k},v^{(0)}_{1},v^{(0)}_{2},\cdots,v^{(0)}_{ns},v^{(1)}_{1},v^{(1)}_{2},\cdots]$ for some $\omega_{1}=v^{(1)}_{1}v^{(1)}_{2}\cdots\in\Gamma$ such that $v^{(0)}_{ns+1}\neq v^{(1)}_{1}$ and

[TABLE]

is irrational, where $u^{(0)}_{s_{0}+1}\neq u^{(1)}_{1}$. By Claim 4.4, $s_{0}>0$.

Next pick a

[TABLE]

and set

[TABLE]

such that $u^{(1)}_{ns+1}\neq u^{(2)}_{1}$ and $f_{\epsilon}(\mu_{2})$ is irrational. Set $\nu_{2}=f_{\epsilon}(\mu_{2})$. Then by Lemmas 2.3 and 3.1,

[TABLE]

Claim 3.4**.**

There exists a positive $s_{1}$ such that

[TABLE]

where $v^{(1)}_{s_{1}+1}\neq v^{(2)}_{1}$.

Proof.

If the continued fraction expansions of $f_{\epsilon}(\mu_{2})$ and $\nu_{1}$ differ before $v^{(1)}_{2}$, by Lemma 2.3 we would have

[TABLE]

This contradicts (8). ∎

Again, pick a

[TABLE]

and set

[TABLE]

such that $v^{(2)}_{ns+1}\neq v^{(3)}_{1}$ and $f^{-1}_{\epsilon}(\nu_{3})$ is irrational. Continuing in this way, we finally get two infinite continued fractions

[TABLE]

and

[TABLE]

where $V^{(2l)}$ and $U^{(2l+1)}$ are words of length $ns$, and $V^{(2l+1)}$ and $U^{(2l)}$ are finite words of elements from $\{B,B+1\}$. Hence $x$ and $y$ satisfy Conditions (i), (ii) and (iii). ∎

Now for each $\epsilon\in(0,\psi]$, choose two infinite continued fractions

[TABLE]

and

[TABLE]

satisfying Conditions of Lemma 3.2. Set

[TABLE]

By Condition (iii) of Lemma 3.2, any subblock of $[\overline{b_{1}\cdots b_{s}}]$ of length $>2ns$ does not occur in $h_{1},h_{2},\cdots$ except for $\underbrace{b_{1},\cdots,b_{s}}_{k}$. Let $I$ be the set of all $i$ satisfying $h_{i+1},\cdots,h_{i+ks}=\underbrace{b_{1},\cdots,b_{s}}_{k}$. For each $\phi\in\Phi$, set

[TABLE]

[TABLE]

and

[TABLE]

Then by (4) and Lemma 3.2, we have

[TABLE]

hence $c_{\alpha}\xi\leq\epsilon.$ By (3) and (4), in order to prove $c_{\alpha}\xi=\epsilon,$ we need to estimate $\tfrac{2|D_{r}-\phi||\tfrac{1}{d_{r}}+\phi^{\sigma}|}{|\phi-\phi^{\sigma}||D_{r}+\tfrac{1}{d_{r}}|}.$ Assume that $\phi=\alpha_{i}$. Then let $m,l$ be the nonnegative integers such that

[TABLE]

and

[TABLE]

Now we divide the estimation into 3 cases

1. (i)

$m+l\leq 3s$

By Lemma 2.2 we have $|D_{r}-\alpha_{i}|\geq\tfrac{1}{72B^{6s+2}}$ and $|\tfrac{1}{d_{r}}-\tfrac{1}{\alpha^{\prime}_{i}}|\geq\tfrac{1}{72B^{6s+4}}$. If $h_{r+1}\leq B$, we have

[TABLE]

If $h_{r+1}>B$, we have

[TABLE]

Hence

[TABLE] 2. (ii)

$3s<m+l\leq 2ns$. In this case let

[TABLE]

where $1\leq h,j\leq s$. By Lemma 2.4, we have

[TABLE]

By Lemma 2.2, we have

[TABLE]

By Lemma 2.3 and (6), we have

[TABLE]

Now combining (11), (12) and (13) yields that

[TABLE] 3. (iii)

$m+l>2ns$. In this case, we have $r-m\in I$ and, by Lemma 2.4,

[TABLE]

The estimation in the case $\phi=\alpha^{\prime}_{i}$ can be dealt with in a similar way. Hence we show that

[TABLE]

This finishes the proof of Theorem 1.1.

4. Hurwitz constant of quadratic Lagrange spectrum

From now on we assume that $v\geq u\geq 1$. In this section, we compute Hurwitz constants of quadratic Lagrange spectrums for real quadratic numbers $[\overline{u,v}]$, $u\geq 9$.

Set $\tau_{1}=[\overline{u,v}]$ and $\tau_{2}=[\overline{v,u}]$. Recall that $\varphi$ is the Golden Ratio

[TABLE]

Set

[TABLE]

A direct calculation shows that

[TABLE]

Lemma 4.1**.**

If $(u,v)\neq(1,1)$,

[TABLE]

Proof.

Set $\varphi^{\prime}=[\underbrace{1,\cdots,1}_{r}],$ and let $\tau=\tau_{1}$ or $\tau_{2}$. If we approximate $\varphi$ by

[TABLE]

where $a_{1}\neq 1$, we need to estimate $\frac{2|\varphi-\alpha|}{|\alpha-\alpha^{\sigma}|}.$ Let $\{\frac{p_{n}}{q_{n}}\}_{n\geq 1}$ be the sequence of convergents of $[a_{1},\cdots,a_{n}]$. Applying the deduction of equation (4) in §2 shows that

[TABLE]

where we can assume that $\varphi^{\prime}$ is arbitrarily closed to $\varphi$ if necessary. Let $\tau^{\prime}=-\tfrac{1}{\tau^{\sigma}}$. Then the right hand side of the above equality simplifies to

[TABLE]

Now we follow the arguments in [8] to treat the case $n=1$ (the case $n=0$ can be reduced to the case $n=1$ by setting $a_{1}=[\tau]$ and replacing $\tau$ with $\tau^{\prime}$). In this case, $q_{n}=p_{n-1}=1$ and $q_{n-1}=0$. We need to estimate

[TABLE]

which is an absolute value of a quadratic form in $p_{1}\in\mathbb{Z}$. The minimal value can only be attained for integers closest to the zeroes of the quadratic form, which are $\varphi-\tau^{-1}$ and $\tau^{\prime}-\varphi^{\prime-1}$. Hence the possible minimal integer points are

[TABLE]

Evaluating $Q$ on these integers we get

[TABLE]

where we require $n=1$ and $\tau\in\{\tau_{1},\tau_{2}\}$ in (16).

Now we treat the case $n\geq 2$ and $a_{n}\neq[\tau^{\prime}]$. In this case, as $a_{1}\neq 1$, we have $p_{i}\geq 2q_{i},$ and hence

[TABLE]

for $i=n,n-1$. (17) still holds for $i=n-2$ because when $n=2$, $p_{n-2}-\varphi q_{n-2}=1$. Thus it suffices to show that

[TABLE]

or

[TABLE]

By (15), when $\varphi^{\prime}$ is sufficiently closed to $\varphi$, we have

[TABLE]

Hence it suffices to show that

[TABLE]

When $a_{n}>[\tau^{\prime}]$, the left hand side of (18) is larger than

[TABLE]

When $a_{n}<[\tau^{\prime}]$, the left hand side of (18) is

[TABLE]

This finishes the proof of the lemma. ∎

For an irrational real number

[TABLE]

set

[TABLE]

and set

[TABLE]

Lemma 4.2**.**

For any $v\geq u\geq 1$, we have

[TABLE]

In particular, if $N_{\xi}=\infty$, we have

[TABLE]

Proof.

Without loss of generality, we can assume that

[TABLE]

is a positive number such that $N_{\xi}$ is very large. Set

[TABLE]

and

[TABLE]

If we approximate $\xi$ by

[TABLE]

where $a_{1}\neq b_{r+1}$ and $\tau\in\{\tau_{1},\tau_{2}\}$, we need to estimate

[TABLE]

or

[TABLE]

where $\{\frac{p_{i}}{q_{i}}\}_{i\geq 1}$ is the sequence of convergents of $[a_{1},\cdots,a_{n}]$. The treatment in the case $n=1$ proceeds exactly as above and implies the lower limit is $\frac{2\vartheta_{u,v}}{(\tau_{1}\tau_{2}+1)}$ when $N_{\xi}\rightarrow\infty$.

Now we treat the case $n\geq 2$ and $a_{n}\neq[\tau^{\prime}]$. When $a_{n}>[\tau^{\prime}]$,

[TABLE]

hence

[TABLE]

When $a_{n}<[\tau^{\prime}]$,

[TABLE]

and

[TABLE]

hence (22) still holds.

As we are only concerned with the the lower limit when $N_{\xi}\rightarrow\infty$, by (22), we can replace (21) with

[TABLE]

Claim 4.3**.**

[TABLE]

Proof.

When $a_{n}>[\tau^{\prime}]$, the left hand side is

[TABLE]

When $a_{n}<[\tau^{\prime}]$, the left hand side is

[TABLE]

∎

It remains to estimate $|(p_{n}-D_{r}q_{n})\tau+(p_{n-1}-D_{r}q_{n-1})|.$ We note that as $a_{1}\neq b_{r+1}$, $(p_{n}-D_{r}q_{n})\tau$ and $p_{n-1}-D_{r}q_{n-1}$ can not have opposite signs.

Claim 4.4**.**

If $n=2$ and $a_{2}\neq 1$, or $n=2$ and $a_{1}>b_{r+1}$, or $n>2$, we have

[TABLE]

when $N_{\xi}\rightarrow\infty$.

Proof.

When $n=2$ and $a_{2}\neq 1$, or $n=2$ and $a_{1}>b_{r+1}$, the left hand side simplifies to

[TABLE]

Now we consider the case $n>2$. If $a_{1}-b_{r+1}\neq-1$, we have

[TABLE]

If $a_{1}-b_{r+1}=-1$, direct computation shows that

[TABLE]

and $q_{n}\geq a_{1}a_{3}\geq 2(a_{3}+2)$ when $N_{\xi}\rightarrow\infty$. Hence, we have

[TABLE]

∎

We note that $p_{n-1}\geq a_{1}$. Now if $n=2$ and $a_{2}\neq 1$, or $n=2$ and $a_{1}>b_{r+1}$, or $n>2$, combining Claim 4.3 and Claim 4.4 implies that

[TABLE]

when $N_{\xi}\rightarrow\infty$.

If $n=2$, $a_{2}=1$ and $a_{1}<b_{r+1}$, (23) simplifies to

[TABLE]

when $N_{\xi}\rightarrow\infty$. This completes the proof.∎

Comparing Lemma 4.1 with Lemma 4.2, we have

[TABLE]

when $u\geq 4$.

We are now in the position to determine the Hurwitz constant of quadratic Lagrange spectrum for real quadratic number $[\overline{u,v}]$, $u\geq 9$.

Proof of Theorem 1.2.

Let

[TABLE]

be an irrational real number not in $\Theta_{\varphi}\bigcup\Theta_{[\overline{u,v}]}$ and let $d_{r}$ and $D_{r}$ be as before. If we approximate $\xi$ by

[TABLE]

we need to verify

[TABLE]

The proof is divided into 4 cases

1. i

There exist infinitely many $r$ such that $b_{r},b_{r+1}>1$.

For such $r$, set $k=[\tau^{\prime}]$. Then the left hand side of (26) is

[TABLE]

which is invariant under the interchange $(\tau^{\prime},D_{r})\leftrightarrow(\tau,d_{r})$. Hence we can assume without loss of generality that $d_{r}\geq D_{r}$. Set $\tau=\tau_{2}$.

Claim 4.5**.**

There exist infinitely many $r$ such that $b_{r},b_{r+1}>1$, and

[TABLE]

Proof.

If $b_{r}\geq 3$, it is easy to check that (27)holds.

If $b_{r}\geq 3$ does not occur infinitely, then, as $d_{r}\geq D_{r}$, either the case $b_{r-1}+1=b_{r}=b_{r+1}=2\geq b_{r+2}$ occurs infinitely or $b_{r}=2$ for sufficiently large $r$. We have in the first case $d_{r}>2+\tfrac{1}{2}$, $D_{r}\geq 2+\tfrac{1}{3}$ in the second case $D_{r},d_{r}\rightarrow 1+\sqrt{2}$. Since $\tau_{1}>9$, in both the cases we can verify that (27) still holds. ∎

Now choose an $r$ of Claim 4.5. If $\tau_{2}\leq 2d_{r}$, $\frac{|\tau_{2}-d_{r}|}{d_{r}}\leq 1.$ Since $\tau_{2}>9$, in this case we have

[TABLE]

If $\tau_{2}>2d_{r}$, by (27), we have,

[TABLE]

This settles the case (1).

If the case (1) is excluded, then the tail of $[b_{0},b_{1},\cdots]$ has the form

[TABLE]

where $b_{c_{i}}>1$. Set $M=\limsup_{r\rightarrow\infty}b_{c_{r}}$. 2. ii

$M\geq v$.

For any $b_{c_{r}}\geq v$, set $k=1$ and $\tau=\tau_{2}$ , and replace $r$ with $c_{r}-1$.

Then the left hand side of (26) is

[TABLE] 3. iii

$M<u-1$.

For any $b_{c_{r}}=M$, set $k=M+1$ and $\tau=\tau_{2}$, and replace $r$ with $c_{r}-1$. Then the left hand side of (26) is

[TABLE] 4. iv

$v>M\geq u-1$.

As $M\geq u-1>8$ and $\tau_{1}>9$, interchanging $\tau_{1}$ and $\tau_{2}$ in (3), we get

[TABLE]

∎