Note on the spectrum of classical and uniform exponents of ...

arXiv:1701.01584v1 [math.NT] 6 Jan 2017

Note on the spectrum of classical and uniform exponents of Diophantine approximation Antoine MARNAT∗ [email protected]

Abstract Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer [12, 13] and results by D. Roy [9, 10], we establish that the spectrum of the 2n exponents of Diophantine approximation in dimension n ≥ 3 is a subset of R2n with non empty interior.

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Introduction

Throughout this paper, the integer n ≥ 1 denotes the dimension of the ambient space, θ = (θ1 , . . . , θn ) denotes an n-tuple of real numbers such that 1, θ1 , . . . , θn are Q-linearly independent. Let d be an integer with 0 ≤ d ≤ n − 1. We define the exponent ωd (θ) (resp. the uniform exponent ω ˆ d (θ)) as the supremum of the real numbers ω for which there exist rational affine subspaces L ⊂ Rn such that dim(L) = d , H(L) ≤ H and H(L)d(θ, L) ≤ H −ω for arbitrarily large real numbers H (resp. for every sufficiently large real number H). Here H(L) denotes the height of L (see [11] for more details), and d(θ, L) = minP ∈L d(θ, P ) is the minimal distance between θ and a point of L. These exponents were introduced originally by M. Laurent [6]. They interpolate between the classical exponents ω(θ) = ωn−1 (θ) and λ(θ) = ω0 (θ) (resp. ω ˆ (θ) = ω ˆ n−1 (θ) and ˆ λ(θ) = ω ˆ 0 (θ)) that were introduced by A. Khintchine [4, 5], V. Jarník [3] and Y. Bugeaud supported by the Austrian Science Fund (FWF), Project F5510-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications” ∗

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and M. Laurent [1, 2]. We have the relations ω0 (θ) ≤ ω1 (θ) ≤ · · · ≤ ωn−1 (θ), ω ˆ 0 (θ) ≤ ω ˆ 1 (θ) ≤ · · · ≤ ω ˆ n−1 (θ), and Minkowski’s First Convex Body Theorem [8] and Mahler’s compound convex bodies theory provide the lower bounds d+1 , for 0 ≤ d ≤ n − 1. n−d These exponents happen to be related. Given k exponents e1 , . . . , ek , we define the spectrum of the exponents (e1 , . . . , ek ) as the subset of Rk described by all k-uples (e1 (θ), . . . , ek (θ)) as θ runs through all points θ = (θ1 , . . . , θn ) ∈ Rn such that 1, θ1 , . . . , θn are Q-linearly independent. ωd (θ) ≥ ω ˆ d (θ) ≥

In [7], the author proved the following theorem. Theorem 1. For every integer n ≥ 3, the n uniform exponents ω ˆ0, . . . , ω ˆ n−1 are algebraically independent. In terms of spectrum, this means that for every integer n ≥ 3, the spectrum of ω ˆ0, . . . , ω ˆ n−1 n is a subset of R with nonempty interior. In this paper, we extend this result as follows. Theorem 2. For every integer n ≥ 3, the spectrum of ω ˆ0, . . . , ω ˆ n−1 , ω0 , . . . , ωn−1 is a subset 2n of R with nonempty interior. We refer the reader to [7] for the notation and the presentation of the parametric geometry of numbers, main tool of the proof.

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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 [7, 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 an open set in the spectrum via Roy’s theorem. We fix the dimension n ≥ 3. Consider the family of parameters A1 = A2 < A3 < · · · < An+1 , B2 < B3 < · · · < Bn , C , D

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satisfying the following properties for 2 ≤ k ≤ n: A1 + A2 + · · · + An+1 = 1 , B2 < D < CA2 ,

(1)

Ak+1 < Bk < Ak+2 , Bk < CAk .

We consider the generalized (n + 1)-system P on the interval [1, C] depending on the previous parameters whose combined graph is given below by Figure 1, where Pk (1) = Ak and Pk (C) = CAk for 1 ≤ k ≤ n + 1.

Pn+1

An+1

Pn

Pn+1 ···

Bk

Ak+1 Bk−1 Ak

A4 B2 A3

CAn+1 CAn

···

···

··· Pk

Pk

Pk+1

CAk

Pk−1

Pk−1

CAk−1

Pk ···

···

···

···

P4

CA2 D

P2 P2 ···

···

···

···

P1

A2 1 δ2,1

δn,2 µn

δk−1,1 δk,1 δk−1,2 δk,2

µk−1

µ1

C

Figure 1: Pattern of the combined graph of P on the fundamental interval [1, C] Conditions (1) are consistent with the graph. From 1 to δn,2 , the combined graph is the same as in [7, §5], then every component Pk from k = n down to k = 2 is successively increasing between µk and µk−1 to readjust to the value CAk . The point which will be most relevant for the proof are labeled with black dots. We extend P to the interval [1, ∞) by self-similarity. This means, P (q) = C m P (C −m q) for all integer m. In view of the value of P and its derivatives at 1 and C, one sees that the

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extension provides a generalized (n + 1)-system on [1, ∞). ˆ n−1 , . . . , W ˆ 0 by [7, Proposition 1] suggests to define 2n quantities Wn−1 , . . . , W0 , W P1 (q) + · · · + Pk (q) 1 := lim sup for 1 ≤ k ≤ n, ˆ q q→+∞ 1 + Wn−k P1 (q) + · · · + Pk (q) 1 := lim inf for 1 ≤ k ≤ n. q→+∞ 1 + Wn−k q Here, self-similarity ensures that the lim sup (resp. lim inf) is in fact the maximum (resp. the minimum) on the interval [1, C[. Note that for 1 ≤ k ≤ n, the function P1 + · · · + Pk has slope 1 on the intervals [1, δk,1 ] and [µk , C[, slope 1/2 on the interval [δk,1 , δk,2 ] and is constant on the interval [δk,2 , µk ]. Therefore the minimum of the function q 7→ q −1 (P1 (q) + · · · + Pk (q)) is reached at µk and its maximum is reached either at δk,1 or at δk,2 , when slope changes from 1 to 1/2 or from 1/2 to 0. Namely, the maximum is reached at δk,1 if P1 (δk,1 ) + · · · + Pk (δk,1 ) 1 ≥ δk,1 2

(2)

and at δk,2 if the lefthand side is ≤ 1/2. We deduce that for 1 ≤ k ≤ n, ˆ n−k = W

Pk+1 (q) + · · · + Pn+1 (q) where q = P1 (q) + · · · + Pk (q)

Wn−k =

Pk+1 (µk ) + · · · + Pn+1 (µk ) . P1 (µk ) + · · · + Pk (µk )

(

δk,1 δk,2

if (2) is satisfied otherwise

,

It is easy to check that the parameters C = 3, A1 = A2 = 2−n , Ak = 2−n+k−2 for 3 ≤ k ≤ n + 1 D=

11 −n 5 2 , Bk = 2−n+k−2 for 2 ≤ k ≤ n + 1 8 4

(3)

satisfy the conditions (1). For this choice of parameters, the lefthand side of inequality (2) is > 1/2 for 1 ≤ k ≤ n − 1 and < 1/2 for k = n. This property remains true for (C, A2 , . . . , An , D, B2 , B3 , . . . , Bn ) in an open neighborhood of the point (3, 2−n , . . . , 2−2 ,

5 11 −n 5 −n 2 , 2 , . . . , 2−2 ) 8 4 4

provided that we set A1 = A2 and An+1 = 1 − (A1 + · · · + An ). In this neighborhood, ˆ 0, . . . , W ˆ n−1 are given by the following rational fractions in the quantities W0 , . . . , Wn−1 , W Q(C, A2 , . . . , An , D, B2 , B3 , . . . , Bn ) :

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1 − (2A2 + A3 + A4 + · · · + An 1 ˆ0 = − 1, W , A2 A2 + (B2 + · · · + Bn−1 ) ˆ n−k = 1 − (2A2 + A3 + A4 + · · · + Ak+1 ) + Bk for 2 ≤ k ≤ n − 1, W A2 + (B2 + · · · + Bk ) C(1 − (2A2 + A3 + A4 + · · · + Ak )) Wn−k = for 2 ≤ k ≤ n, A2 + B2 + · · · + Bk D + C(1 − 2A2 ) . Wn−1 = A2 ˆ n−1 = W

ˆ 0, . . . , W ˆ n−1 come from a generalized (n + 1)-system P , Roy’s TheSince W0 , . . . , Wn−1 , W ˆ k and ωk (θ) = Wk orem provides the existence of a point θ in Rn such that ω ˆ k (θ) = W for every 0 ≤ k ≤ n − 1. Therefore, to prove Theorem 2 it is sufficient to show that the ˆ 0, . . . , W ˆ n−1 ∈ Q(C, A2 , A3 , . . . , An , D, B2 , B3 , . . . , Bn ) are rational fractions W0 , . . . , Wn−1 , W algebraically independent. ˆ n−1 only depends on A2 . Therefore, it First, note that only Wn−1 depends on D and W is enough to prove that the 2n − 2 other rational fraction are algebraically independent. For the calculation, it is convenient to successively make the following two change of variables. First, we set Mk := 1 −

k X

Ai , Nk := A1 +

i=1

k X

Bi for 2 ≤ k ≤ n + 1.

i=2

Note that Mn+1 = 1. We get the formulae

Mn , Nn−1 CMk for 2 ≤ k ≤ n, = Nk Mk+1 − Nk−1 = 1+ for 2 ≤ k ≤ n − 1. Nk

ˆ0 = W Wn−k ˆ n−k W Then, we set Uk :=

Mk Mk+1 and Vk := for 2 ≤ k ≤ n + 1. Nk Nk

Note that V1 = 2 getting the formulae ˆ 0 = Vn−1 , W Wn−k = CUk for 2 ≤ k ≤ n ˆ n−k = 1 + Vk − Uk for 2 ≤ k ≤ n − 1. W Vk−1

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Hence, the 2n − 2 independent parameters C, A3 , · · · , An , B2 , · · · , Bn provide the 2n − 2 independent parameters C, U2 , . . . , Un , V2 , . . . , Vn−1 . Thus, it is sufficient to show that the ˆ 0, . . . , W ˆ n−2 ∈ Q(C, U2 , U3 , . . . , Un , V2 , V3 , . . . , Vn−1 ) are rational fractions W0 , . . . , Wn−2 , W algebraically independent. Suppose that there exists an irreducible polynomial R ∈ Q(X1 , . . . , X2n−2 ) such that ˆ 0, . . . , W ˆ n−2 , W0 , . . . , Wn−2 = 0. R W 



Specializing C in 1, we obtain

U2 Un−1 , . . . , V2 + 1 − , Un , . . . , U2 R Vn−1 , Vn−1 + 1 − Vn−2 2

!

=0

where the 2n − 3 last rational fractions generate the field Q(U2 , . . . , Un , V2 , . . . , Vn−1 ). Therefore, they are algebraically independent. We investigate their relation with the first coordinate, that will provides R. Observe that for 2 ≤ k ≤ n − 1, ˆ n−k = 1 + Vk − Uk W Vk−1 provides the relation ˆ n−k − 1 + Wn−k . Vk = W Vk−1 Therefore, initializing with ˆ n−2 + 1 − Wn−2 , V2 = W V1 1 − 2A2 is a constant, we can compute by recurrence Vk in terms of the raA2 ˆ n−2 , . . . , W ˆ n−k , Wn−2 , . . . , Wn−k and the constant V1 and observe that, if tional fractions W not specialized, it is not homogeneous in C. For k = n − 1, we have the additional relation ˆ 0 providing a polynomial relation. Up to multiplication by a nonzero constant, this Vn−1 = W polynomial is R. The contradiction follows from R not being homogeneous in C. where V1 =

References [1] Yann Bugeaud and Michel Laurent. On exponents of homogeneous and inhomogeneous diophantine approximation. Moscow Math. J., 5:747–766, 2005. [2] Yann Bugeaud and Michel Laurent. Exponents of diophantine approximation. Diophantine Geometry Proceedings, 4:101–121, 2007.

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[3] Vojtěch Jarník. Zum khintchineschen "Übertragungssatz". Trav. Inst. Math. Tbilissi, 3:193–212, 1938. [4] Alexander Ya. Khinchin. Über eine klasse linearer diophantischer approximationen. Rend. Circ. Mat. Palermo 50, pages 170–195, 1926. [5] Alexander Ya. Khinchin. Zur metrischen theorie der diophantischen approximationen. Math.Z., 24:706–714, 1926. [6] Michel Laurent. On transfer inequalities in Diophantine approximation. In Analytic number theory, pages 306–314. Cambridge Univ. Press, Cambridge, 2009. [7] A. Marnat. About Jarník’s type relation in higher dimension. ArXiv e-prints, October 2015. [8] Hermann Minkowski. Geometrie der Zahlen. Bibliotheca Mathematica Teubneriana, Band 40. Johnson Reprint Corp., New York-London, 1968. [9] Damien Roy. On Schmidt and Summerer parametric geometry of numbers. Ann. of Math., 182:739–786, 2015. [10] Damien Roy. Spectrum of the exponents of best rational approximation. Math. Z., 283:143–155, 2016. [11] Wolfgang M. Schmidt. On heights of algebraic subspaces and diophantine approximations. Ann. of Math. (2), 85:430–472, 1967. [12] Wolfgang M. Schmidt and Leonhard Summerer. Parametric geometry of numbers and applications. Acta Arithmetica, 140(1):67–91, 2009. [13] Wolfgang M. Schmidt and Leonhard Summerer. Diophantine approximation and parametric geometry of numbers. Monatsch. Math., 169(1):51–104, 2013.

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