Note on the spectrum of classical and uniform exponents of Diophantine approximation

TL;DR

This paper explores the range of classical and uniform exponents in Diophantine approximation for dimensions three and higher, showing that their spectrum has a rich structure with non-empty interior.

Contribution

It establishes that the spectrum of 2n exponents in dimension n≥3 is a subset of R^{2n} with non-empty interior, using Parametric Geometry of Numbers and recent results.

Findings

Spectrum of 2n exponents has non-empty interior for n≥3

Utilizes Parametric Geometry of Numbers and recent theoretical results

Extends understanding of Diophantine approximation exponents in higher dimensions

Abstract

Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we establish that the spectrum of the $2n$ exponents of Diophantine approximation in dimension $n\geq3$ is a subset of $\mathbb{R}^{2n}$ with non empty interior.

Equations

$$\dim (L)=d,H(L)\le H\text{ and }H(L)d(\mathbf{\theta },L)\le H^{-\omega }$$

$${\omega }_0(\mathbf{\theta })\le {\omega }_1(\mathbf{\theta })\le \cdots \le {\omega }_{n-1}(\mathbf{\theta }),$$

$${\hat{\omega }}_0(\mathbf{\theta })\le {\hat{\omega }}_1(\mathbf{\theta })\le \cdots \le {\hat{\omega }}_{n-1}(\mathbf{\theta }),$$

$${\omega }_d(\mathbf{\theta })\ge {\hat{\omega }}_d(\mathbf{\theta })\ge \frac{d+1}{n-d},\text{ for }0\le d\le n-1.$$

$${\hat{\omega }}_0+1/{\hat{\omega }}_1=1,$$

$$A_1=A_2<A_3<\cdots <A_{n+1},B_2<B_3<\cdots <B_n,C,D$$

$$\begin{array}{rl}{A}_1+A_2+\cdots +A_{n+1}& =1,B_2<D<C{A}_2,\\ A_{k+1}<B_k<A_{k+2}& ,B_k<C{A}_k,\end{array}$$

$$P_k(1)=A_k\text{ and }{P}_k(C)=C{A}_k\text{ for }1\le k\le n+1.$$

$$\frac{1}{1+{\hat{W}}_{n-k}}$$

$$\frac{1}{1+W_{n-k}}$$

$$\frac{P_1({\delta }_{k,1})+\cdots +P_k({\delta }_{k,1})}{{\delta }_{k,1}}\ge \frac{1}{2}$$

$${\hat{W}}_{n-k}$$

$$W_{n-k}$$

$$\begin{array}{rl}C=3,A_1=A_2=2^{-n},A_k& =2^{-n+k-2}\text{ for }3\le k\le n+1\\ D=\frac{11}{8}{2}^{-n+1},B_k& =\frac{5}{4}{2}^{-n+k-1}\text{ for }2\le k\le n\end{array}$$

$$(3,2^{-n},\dots ,2^{-2},\frac{11}{8}{2}^{-n+1},\frac{5}{2}{2}^{-n},\dots ,\frac{5}{2}{2}^{-2})$$

$$\begin{array}[]{ll}\hat{W}_{n-1}=\cfrac{1}{A_{2}}-1,&\hat{W}_{0}=\cfrac{1-(2A_{2}+A_{3}+A_{4}+\cdots+A_{n})}{A_{2}+(B_{2}+\cdots+B_{n-1})},\\ \hat{W}_{n-k}=\cfrac{1-(2A_{2}+A_{3}+A_{4}+\cdots+A_{k+1})+B_{k}}{A_{2}+(B_{2}+\cdots+B_{k})}&\textrm{ for }2\leq k\leq n-1,\\ W_{n-k}=\cfrac{C(1-(2A_{2}+A_{3}+A_{4}+\cdots+A_{k}))}{A_{2}+B_{2}+\cdots+B_{k}}&\textrm{ for }2\leq k\leq n,\\ W_{n-1}=\cfrac{D+C(1-2A_{2})}{A_{2}}.\end{array}$$

$$M_k$$

$$N_k$$

$${\hat{W}}_0$$

$$W_{n-k}$$

$${\hat{W}}_{n-k}$$

$$U_k:=\frac{M_k}{N_k}\text{ and }{V}_k:=\frac{M_{k+1}}{N_k}\text{ for }2\le k\le n,$$

$${\hat{W}}_0$$

$$W_{n-k}$$

$${\hat{W}}_{n-k}$$

$$R\left({\hat{W}}_0,\dots ,{\hat{W}}_{n-2},W_0,\dots ,W_{n-2}\right)=0.$$

$$R\left(V_{n-1},V_{n-1}+1-\frac{U_{n-1}}{V_{n-2}},\dots ,V_2+1-\frac{U_2}{V_1},U_n,\dots ,U_2\right)=0$$

$${\hat{W}}_{n-k}=1+V_k-\frac{U_k}{V_{k-1}}$$

$$V_k={\hat{W}}_{n-k}-1+\frac{W_{n-k}}{V_{k-1}}.$$

$${\hat{W}}_0=V_{n-1}=({\hat{W}}_1-1)+\frac{W_1}{V_{n-2}}=f_0+\underset{k=1}{\stackrel{n-2}{\mathrm{K}}}\frac{e_k}{f_k}$$

$$\left\{\begin{array}[]{clr}e_{k}&=W_{k}&\textrm{ for }1\leq k\leq n-2\\ f_{k}&=\hat{W}_{k+1}-1&\textrm{ for }0\leq k\leq n-3\\ f_{n-2}&=V_{1}=\cfrac{1-2A_{2}}{A_{2}}&\end{array}\right.$$

$$f_0+\underset{k=1}{\stackrel{n-2}{\mathrm{K}}}\frac{e_k}{f_k}=f_0+\frac{e_1}{f_1+\frac{e_2}{f_2+\frac{\ddots }{f_{n-2}}}}$$

$$\tilde{R}=F_{n-2}{\hat{W}}_0-E_{n-2}$$

$$R=\tilde{R}Q+P$$

Full text

1 Introduction

Throughout this paper, the integer $n\geq 1$ denotes the dimension of the ambient space $\mathbb{R}^{n}$ endowed with its Euclidean norm and $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{n})$ denotes an $n$-tuple of real numbers such that $1,\theta_{1},\ldots,\theta_{n}$ are $\mathbb{Q}$-linearly independent.

Let $d$ be an integer with $0\leq d\leq n-1$. We define the exponent ${\omega}_{d}(\boldsymbol{\theta})$ (resp. the uniform exponent $\hat{\omega}_{d}(\boldsymbol{\theta})$) as the supremum of the real numbers $\omega$ for which there exist rational affine subspaces $L\subset\mathbb{R}^{n}$ such that

[TABLE]

for arbitrarily large real numbers $H$ (resp. for every sufficiently large real number $H$). Here $H(L)$ denotes the exponential height of $L$ (see [12] for more details), and $d(\boldsymbol{\theta},L)=\min_{P\in L}d(\boldsymbol{\theta},P)$ is the minimal distance between $\boldsymbol{\theta}$ and a point of $L$. Note that this definition is independent of the choice of a norm on $\mathbb{R}^{n}$.

These exponents were introduced originally by M. Laurent [7]. They interpolate between the classical exponents $\omega(\boldsymbol{\theta})=\omega_{n-1}(\boldsymbol{\theta})$ and $\lambda(\boldsymbol{\theta})=\omega_{0}(\boldsymbol{\theta})$ (resp. $\hat{\omega}(\boldsymbol{\theta})=\hat{\omega}_{n-1}(\boldsymbol{\theta})$ and $\hat{\lambda}(\boldsymbol{\theta})=\hat{\omega}_{0}(\boldsymbol{\theta})$) that were introduced by A. Khintchine [4, 5], V. Jarník [3] and Y. Bugeaud and M. Laurent [1, 2].

We have the relations

[TABLE]

[TABLE]

and Minkowski’s First Convex Body Theorem [9] and Mahler’s compound convex bodies theory provide the lower bounds

[TABLE]

These $2n$ exponents happen to be related as was first noticed by Khinchin with his transference theorem [5]. We use the following notion of spectrum to study more general transfers. Given $k$ exponents $e_{1},\ldots,e_{k}$, we define the spectrum of the exponents $(e_{1},\ldots,e_{k})$ as the subset of $\mathbb{R}^{k}$ described by all $k$-uples $(e_{1}(\boldsymbol{\theta}),\ldots,e_{k}(\boldsymbol{\theta}))$ as $\boldsymbol{\theta}$ runs through all points $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{n})\in\mathbb{R}^{n}$ such that $1,\theta_{1},\ldots,\theta_{n}$ are $\mathbb{Q}$-linearly independent.

In [8], the author proved the following theorem.

Theorem 1**.**

For every integer $n\geq 3$, the $n$ uniform exponents $\hat{\omega}_{0},\ldots,\hat{\omega}_{n-1}$ are algebraically independent.

Using the same construction, it is even possible to show that for every integer $n\geq 3$, the spectrum of $\hat{\omega}_{0},\ldots,\hat{\omega}_{n-1}$ is a subset of $\mathbb{R}^{n}$ with non empty interior. In this paper, we extend this result as follows.

Theorem 2**.**

For every integer $n\geq 3$, the $2n$ exponents $\hat{\omega}_{0},\ldots,\hat{\omega}_{n-1},\omega_{0},\ldots,\omega_{n-1}$ are algebraically independent.

In dimension $n=2$, the spectrum is fully described in [6]:

Theorem 3** (Laurent, 2009).**

In dimension $2$, the spectrum of the four exponents $\omega_{0},\omega_{1},\hat{\omega}_{0},\hat{\omega}_{1}$ is described by the inequalities

[TABLE]

When $\hat{\omega}_{1}<\omega_{1}=+\infty$ we have to understand these relations as $\hat{\omega}_{1}-1\leq\omega_{0}\leq+\infty$ and when $\hat{\omega}_{1}=+\infty$, the set of constraints should be interpreted as ${\omega}_{0}=\omega_{1}=+\infty$ and $\hat{\omega}_{0}=1$.

The first equality, relating the two uniform exponents, is known as Jarník’s relation [3] and breaks the algebraic independence. Note that this sharpens previously mentioned relations. In dimension $n=1$ the uniform exponent is always equal to $1$.

We refer the reader to [8, §2] for the notation and the presentation of the parametric geometry of numbers, main tool of the proof. We mainly use the notation introduced by D. Roy in [10, 11] which is essentially dual to the one of W. M. Schmidt and L. Summerer [13, 14].

2 Proof of the main Theorem 2

To prove Theorem 2, we place ourselves in the context of parametric geometry of numbers. We fully use Roy’s theorem [8, Theorem 5] that reduces the study of spectra of Diophantine approximation to the study of the combinatorial properties of generalized $n$-systems. We construct explicitly a family of generalized $(n+1)$-systems with $2n$ parameters, which provides the algebraic independence in the spectrum via Roy’s theorem.

We fix the dimension $n\geq 3$. Consider any family of positive parameters

[TABLE]

satisfying the following properties for $2\leq k\leq n$:

[TABLE]

where $A_{n+2}=\infty$.

We consider the generalized $(n+1)$-system $\boldsymbol{P}$ on the interval $[1,C]$ depending on the previous parameters whose combined graph is given below by Figure 1, where

[TABLE]

Conditions (1) are consistent with the graph. On each interval between two consecutive division points, there is only one line segment with non zero slope. This line segment has slope $1$ on the intervals $[1,\delta_{2,1}]$, $[\delta_{k-1,2},\delta_{k,1}]$ for $3\leq k\leq n$, and $[\mu_{k},\mu_{k-1}]$ for $n\geq k\geq 1$, and has slope $1/2$ on the interval $[\mu_{0},C]$ and $[\delta_{k,1},\delta_{k,2}]$ for $3\leq k\leq n$ , where the two components $P_{k}$ and $P_{k+1}$ coincide. We have $3n+1$ division points $1$, $C$, $\delta_{k,1}$ and $\delta_{k,2}$ for $2\leq k\leq n$ and $\mu_{l}$ for $n+1\geq l\geq 0$. They are all ordinary division points except $\mu_{k}$ for $1\leq k\leq n$ which are switch points.

The points which will be most relevant for the proof are labeled with black dots. Note that from $1$ to $\delta_{n,2}$, the combined graph is the same as in [8, §5].

We extend $\boldsymbol{P}$ to the interval $[1,\infty)$ by self-similarity. This means, $\boldsymbol{P}(q)=C^{m}\boldsymbol{P}(C^{-m}q)$ for all integers $m$. In view of the value of $\boldsymbol{P}$ and its derivative at $1$ and $C$, one sees that the extension provides a generalized $(n+1)$-system on $[1,\infty)$.

The relation between exponents and $n$-systems [8, Proposition 1] suggests to define $2n$ quantities $W_{n-1},\ldots,W_{0},\hat{W}_{n-1},\ldots,\hat{W}_{0}$ by

[TABLE]

Indeed with this setting, Roy’s Theorem provides the existence of a point $\boldsymbol{\theta}$ in $\mathbb{R}^{n}$ such that $\hat{\omega}_{k}(\boldsymbol{\theta})=\hat{W}_{k}$ and $\omega_{k}(\boldsymbol{\theta})=W_{k}$ for every $0\leq k\leq n-1$.

Here, self-similarity ensures that the $\limsup$ (resp. $\liminf$) is in fact the maximum (resp. the minimum) on the interval $[1,C[$. Note that for $1\leq k\leq n$, the function $P_{1}+\cdots+P_{k}$ has slope $1$ on the intervals $[1,\delta_{k,1}]$ and $[\mu_{k},C[$, slope $1/2$ on the interval $[\delta_{k,1},\delta_{k,2}]$ and is constant on the interval $[\delta_{k,2},\mu_{k}]$. Therefore the minimum of the function $q\mapsto q^{-1}(P_{1}(q)+\cdots+P_{k}(q))$ is reached at $\mu_{k}$ and its maximum is reached either at $\delta_{k,1}$ or at $\delta_{k,2}$, when slope changes from $1$ to $1/2$ or from $1/2$ to [math]. Namely, the maximum is reached at $\delta_{k,1}$ if

[TABLE]

and at $\delta_{k,2}$ if the lefthand side is $\leq 1/2$. We deduce that for $1\leq k\leq n$,

[TABLE]

It is easy to check that the parameters

[TABLE]

satisfy the conditions (1). For this choice of parameters, the lefthand side of inequality (2) is $>1/2$ for $1\leq k\leq n-1$ and $<1/2$ for $k=n$. This property remains true for $(C,A_{2},\ldots,A_{n},D,B_{2},B_{3},\ldots,B_{n})$ in an open neighborhood of the point

[TABLE]

provided that we set $A_{1}=A_{2}$ and $A_{n+1}=1-(A_{1}+\cdots+A_{n})$. In this neighborhood, the quantities $W_{0},\ldots,W_{n-1},\hat{W}_{0},\ldots,\hat{W}_{n-1}$ are given by the following rational fractions in $\mathbb{Q}(C,A_{2},\ldots,A_{n},D,B_{2},B_{3},\ldots,B_{n})$ :

[TABLE]

Since $W_{0},\ldots,W_{n-1},\hat{W}_{0},\ldots,\hat{W}_{n-1}$ come from a generalized $(n+1)$-system $\boldsymbol{P}$, Roy’s Theorem provides the existence of a point $\boldsymbol{\theta}$ in $\mathbb{R}^{n}$ such that $\hat{\omega}_{k}(\boldsymbol{\theta})=\hat{W}_{k}$ and $\omega_{k}(\boldsymbol{\theta})=W_{k}$ for every $0\leq k\leq n-1$. Therefore, to prove Theorem 2 it is sufficient to show that the rational fractions $W_{0},\ldots,W_{n-1},\hat{W}_{0},\ldots,\hat{W}_{n-1}\in\mathbb{Q}(C,A_{2},A_{3},\ldots,A_{n},D,B_{2},B_{3},\ldots,B_{n})$ are algebraically independent.

First, note that only $W_{n-1}$ depends on $D$ and $\hat{W}_{n-1}$ only depends on $A_{2}$. Therefore, it is enough to prove that the $2n-2$ other rational fractions are algebraically independent over $\mathbb{Q}(A_{2})$. For the calculation, it is convenient to successively make the following two changes of variables. First, we set

[TABLE]

Note that $M_{n+1}=0$ and $N_{1}=A_{1}$. We get the formulae

[TABLE]

Then, we set

[TABLE]

and $V_{1}=\frac{1-2A_{2}}{A_{2}}$ getting the formulae

[TABLE]

Hence, the $2n-2$ independent parameters $C,A_{3},\cdots,A_{n},B_{2},\cdots,B_{n}$ provide the $2n-2$ independent parameters $C,U_{2},\ldots,U_{n},V_{2},\ldots,V_{n-1}$. Thus, it is sufficient to show that the rational fractions $W_{0},\ldots,W_{n-2},\hat{W}_{0},\ldots,\hat{W}_{n-2}\in\mathbb{Q}(A_{2})(C,U_{2},U_{3},\ldots,U_{n},V_{2},V_{3},\ldots,V_{n-1})$ are algebraically independent over $\mathbb{Q}(A_{2})$.

Suppose that there exists an irreducible polynomial $R\in\mathbb{Q}(A_{2})[X_{1},\ldots,X_{2n-2}]$ such that

[TABLE]

Specializing $C$ in $1$, we obtain

[TABLE]

where the $2n-3$ last rational fractions generate the field $\mathbb{Q}(A_{2})(U_{2},\ldots,U_{n},V_{2},\ldots,V_{n-1})$ over $\mathbb{Q}(A_{2})$. Therefore, they are algebraically independent. We investigate their relation with the first coordinate, that will provide information on $R$. Observe that for $2\leq k\leq n-1$,

[TABLE]

provide the relation

[TABLE]

Since $\hat{W}_{0}=V_{n-1}$, we can compute by finite induction

[TABLE]

where

[TABLE]

and

[TABLE]

is Gauss’ notation for a (finite) generalized continued fraction. Denote by $\left(\cfrac{E_{k}}{F_{k}}\right)_{k=0}^{n-2}$ the finite sequence of its convergents.

We set

[TABLE]

where $F_{n-2}$ and $E_{n-2}$ are seen as polynomials in $\mathbb{Q}(A_{2})[W_{0},\ldots,W_{n-2},\hat{W}_{0},\ldots,\hat{W}_{n-2}].$ Note that $F_{n-2}$ and $E_{n-2}$ do not depend on $\hat{W}_{0}$ since none of the $(e_{k})_{1\leq k\leq n-2}$ and $(f_{k})_{0\leq k\leq n-2}$ do. Hence, $\tilde{R}$ is a polynomial of degree $1$ with respect to $\hat{W}_{0}$. Writing the Euclidean division of $R$ by $\tilde{R}$ in $\mathbb{Q}(A_{2},\hat{W}_{1},\ldots,\hat{W}_{n-2},W_{0},\ldots,W_{n-2})[\hat{W}_{0}]$ we get

[TABLE]

with $\deg_{\hat{W}_{0}}(P)=0$. Hence $P$ can be seen as a polynomial in the $2n-3$ variables $\hat{W}_{1},\ldots,\hat{W}_{n-2},W_{0},\ldots,W_{n-2}$ over $\mathbb{Q}(A_{2})$. The latter are algebraically independent over $\mathbb{Q}(A_{2})$ because their specializations at $C=1$ are. We deduce that $P=0$, and by irreducibility of $R$, the polynomial $Q$ is a constant:

[TABLE]

with $\alpha\in\mathbb{Q}(A_{2})$.

Specializing $C$ in [math], we obtain

[TABLE]

where the $n-1$ non zero rational fractions generate the field $\mathbb{Q}(V_{1})(U_{3},\ldots,U_{n-1})(V_{n-1},V_{n-2},\ldots,V_{2},U_{2})$ over $\mathbb{Q}(V_{1})(U_{3},\ldots,U_{n-1})$. Therefore, they are algebraically independent over $\mathbb{Q}(A_{2})=\mathbb{Q}(V_{1})$. We deduce that the constant monomial of $R$ seen in $\mathbb{Q}(A_{2},\hat{W}_{0},\ldots,\hat{W}_{n-2})[{W}_{0},\ldots,{W}_{n-2}]$ should be zero.

We now compute the constant monomial of $F_{n-2}\hat{W}_{0}-E_{n-2}$ seen in $\mathbb{Q}(A_{2},\hat{W}_{0},\ldots,\hat{W}_{n-2})[{W}_{0},\ldots,{W}_{n-2}]$. We use the classical recurrence formulae for the convergents

[TABLE]

to compute the constant term of $E_{n-2}$ and $F_{n-2}$ to be

[TABLE]

respectively. Thus the constant monomial of $F_{n-2}\hat{W}_{0}-E_{n-2}$ seen in $\mathbb{Q}(A_{2},\hat{W}_{0},\ldots,\hat{W}_{n-2})[{W}_{0},\ldots,{W}_{n-2}]$ is

[TABLE]

The fact that $\hat{W}_{k+1}\neq 1$ and $\hat{W}_{0}+1\neq\hat{W}_{1}$ induces that this constant monomial is non zero. Hence $\alpha$ and $R$ are zero.

This proves the algebraic independence of the $2n$ exponents. ∎