24 Septembre 2002
SIMULTANEOUS RATIONAL APPROXIMATION TO
THE SUCCESSIVE POWERS OF A REAL NUMBER
by Michel LAURENT (*)
Abstract. Let n be an integer ≥1 and let θ be a real number which is not an algebraic number of degree ≤n/2.
We show that there exist >0 and arbitrary large real numbers X such that the system of linear inequalities
|x0 |≤X and |x0 θ j −xj |≤X −1/n/2 for 1≤j≤n, has only the zero solution in rational integers x0 ,...,xn . This
result refines a similar statement due to H. Davenport and W. M. Schmidt, where the upper integer part n/2 is
replaced everywhere by the integer part n/2. As a corollary, we improve slightly the exponent of approximation
to θ by algebraic integers of degree n+1 over Q obtained by these authors.
1. Introduction and results.
Let θ be a real number and let n be an integer ≥ 1. We say that a positive real number
λ is an exponent of simultaneous approximation to the numbers θ, . . . , θn if for all > 0
and for all positive real number X sufficiently large in term of , the inequalities
|x0 | ≤ X
and
max |x0 θj − xj | ≤ X −λ
1≤j≤n
have a non zero integer solution x = (x0 , . . . , xn ). The exponent λ measures the degree of
singularity of the system of linear forms θX0 − X1 , . . . , θn X0 − Xn in the sense of Chapter
5.7 of [2]. The notion of singularity introduced by J. W. S. Cassels corresponds in fact to
the exponent λ = 1/n. Notice however that D. Roy has constructed in [5] transcendental
√
numbers θ which are singular for n = 2 and for any exponent λ < (−1 + 5)/2 0.618.
The goal of this paper is to sharpen slightly the upper bound for such an exponent λ of
simultaneous approximation obtained by H. Davenport and W. M. Schmidt in Theorem
2a of [3].
In the case n = 1, it is easily seen that each λ < 1 is an exponent of approximation,
while λ = 1 is not an exponent of approximation for any irrational number θ. It follows
immediately that an exponent of simultaneous approximation λ is < 1 for any n ≥ 1,
assuming only that θ is irrational. Our result, as well as Davenport-Schmidt Theorem,
refines this elementary observation.
(*)
Institut de Mathématiques de Luminy, CNRS.
1
M. Laurent
For any positive real number x, denote by
x
if x is an integer
x =
x
+ 1 if x is not an integer
the upper integer part of x, where x
stands for its usual integer part.
Theorem. Let n be an integer ≥ 1 and let θ be a real number which is not an algebraic
number of degree ≤ n/2. Then for any exponent λ of simultaneous approximation to
θ, . . . , θn we have the upper bound
1
2/n
if n is even
=
.
λ<
2/(n + 1) if n is odd
n/2
1
is not
Notice that an equivalent formulation of our Theorem asserts that λ = n/2
n
an exponent of simultaneous approximation to θ, . . . , θ . On the other hand, Dirichlet box
principle implies that any positive real number
λ<
1
dimQ (Q + Qθ + · · · + Qθn ) − 1
is such an exponent. Thus the Theorem is optimal whenever θ is a real algebraic number
of degree n/2 + 1 over Q.
H. Davenport and W. M. Schmidt have proven exactly the same type of statement
with the upper integer part n/2 replaced everywhere by the integer part n/2
. The two
results are thus identical when n is even, while our upper bound is sharper for odd values
of n. Moreover the assertion in the even case turns out to be a formal consequence of the
odd one, noting that an exponent of simultaneous approximation to θ, . . . , θn is obviously
an exponent of simultaneous approximation to θ, . . . , θn−1 and that (n − 1)/2 = n/2
when n is even. Notice also that for n = 3, our result includes Theorem 4a of [3].
Simultaneous approximation assertions of the type of our Theorem, or those of [3],
imply automatically results of approximation by algebraic integers, using the transference
principle of [3] combined with the ideas of [1]. We obtain the following refinement of
Theorem 2 from [3].
Corollary. Let n be an integer ≥ 2 and let θ be a real number which is not an algebraic
number of degree ≤ (n − 1)/2. Then there are infinitely many algebraic integers α of
degree n which satisfy
0 < |θ − α| H(α)−(n+1)/2 ,
where H(α) denotes the usual height of the algebraic number α.
2
Simultaneous approximation of the successive powers of a real number
2. Some tools for the proof.
The main ingredient in the proof of our Theorem is the following lemma on linear
recursions due to H. Davenport and W. M. Schmidt. We reproduce here their statement
(Theorem 3 from [3]). See also the references [4] and [6], which contain various applications
and new insights on the result.
Lemma 1. Let h ≤ m be positive integers and let z0 , z1 , . . . , zm be elements of Zh which
generate over Q the whole space Qh , and which satisfy the linear recurrence relations
a0 zj + a1 zj+1 + · · · + ah zj+h = 0
(j = 0, . . . , m − h)
for some coprime integer coefficients a0 , . . . , ah . Then
max(|a0 |, . . . , |ah |) Z 1/(m+1−h)
where Z denotes the supremum of the absolute values of the minors of order h extracted
from the h × (m + 1) matrix (z0 , . . . , zm ), and where the constant involved in the symbol
depends only upon h and m.
We shall also use the following elementary result.
Lemma 2. Let a0 , . . . , a and b0 , . . . , b be two sequences of + 1 complex numbers. Let
m be a positive integer. Suppose that the two vector spaces of C+m generated by the m
rows of the m × ( + m) matrices





a0
a1
...
a
0
..
.
a0
..
.
a1
..
.
0
...
0
. . . a
..
.
a0 a1
0
...
..
.
..
.
...

0
.. 
. 

0
a

and




b0
b1
...
b
0
..
.
b0
..
.
b1
..
.
0
...
0
. . . b
..
.
b 0 b1
0
...
..
.
..
.
...

0
.. 
. 

0
b
are equal. Then there exists a non zero complex number ρ such that ak = ρbk for 0 ≤ k ≤ .
Proof. Consider the two polynomials
A = a0 + · · · + a X and B = b0 + · · · + b X .
By hypothesis, we have the equality of vector spaces A · C[X]m−1 = B · C[X]m−1 in
C[X]+m−1 , where C[X]k stands for the C-vector space of polynomials with degree ≤ k.
We deduce immediately that the A = ρB for some non zero ρ ∈ C.
3
M. Laurent
3. Sequences of best approximations.
From now, we begin the proof of our Theorem, following closely the lines of [3] with
the exception of Section 5 which differs.
As was observed in the introduction, we may restrict to odd values of n. Suppose on
the contrary that
1
2
λ=
=
n/2
n+1
is an exponent of simultaneous approximation to the numbers θ, . . . , θn . Then for all
positive and all sufficiently large X, there exists a non zero integer point x = (x0 , . . . , xn )
such that
|x0 | ≤ X
(1)
and L(x) := max |x0 θj − xj | ≤ X −λ .
1≤j≤n
Assumption (1) will imply ultimately that θ is an algebraic number of degree ≤ (n + 1)/2,
provided should have been chosen sufficiently small in term of θ and n. This will be
implicitely supposed in all the forthcoming assertions. The constants involved in the
symbols will depend only on n and θ.
First we construct inductively a sequence of non zero integer points
xi = (xi,0 , . . . , xi,n )
(i ≥ 1)
called best approximations, which satisfy the following defining properties. Set
|xi,0 | = Xi
and Li = L(xi ).
Then we have
1 ≤ X 1 < X2 < · · ·
and L1 > L2 > · · ·
,
and L(x) ≥ Li for all non zero integer point x with |x0 | < Xi+1 . Notice that the sequence
(xi )i≥1 is determined by its initial element x1 which is a minimal point in the sense of [3].
We fix such a sequence (xi )i≥1 and will be concerned with properties which are valid for
large i, meaning that Xi is greater than some value X(θ, n, ). It follows from (1) that, for
large i, we have the upper bound
−λ
.
Li ≤ Xi+1
(2)
Reset tentatively xi = (y0 , . . . , yn ) for the coordinates of the point xi , and denote by
hi the smallest positive integer h such that the (h + 1) × (n − h + 1) Hankel matrix (*)


y0 . . . yn−h
 ..
.. 
(3)
 .
. 
yh
...
yn
should have a rank ≤ h.
(*)
A matrix Y will be called an Hankel matrix if Y =(yi+j ) 0≤i≤p for some sequence y0 ,...,yp+q of complex
0≤j≤q
numbers.
4
Simultaneous approximation of the successive powers of a real number
Lemma 3. For any i ≥ 1, we have the upper bound
hi ≤
n+1
.
2
Proof. We have to prove that the rank of the matrix (3) is ≤ h when h = (n + 1)/2. It is
obvious since then (3) is an (h + 1) × h matrix.
4. Construction of polynomials.
As in the preceding section, denote by (y0 , . . . , yn ) the coordinates of the point xi . By
definition of hi , the Hankel matrix


y0 . . . yn−hi
 ..
.. 
(4)
 .
. 
yhi
(i)
...
yn
(i)
has rank ≤ hi . Let a0 , . . . , ahi be the coefficients of some non trivial linear relation
(i)
(i)
between the hi + 1 rows of the matrix (4). We may suppose that a0 , . . . , ahi are coprime
integers. Thus we have the recursion formulas
(i)
(i)
a0 yj + · · · + ahi yj+hi = 0
(5)
(0 ≤ j ≤ n − hi ).
Set
Pi (X) =
(i)
a0
+ ··· +
(i)
ahi X hi
and H(Pi ) = max
(i)
(i)
|a0 |, . . . , |ahi |
.
Lemma 4. For any sufficiently large i, we have the upper bounds
H(Pi ) Xiλ
−λ
and |Pi (θ)| Xiλ−1 Xi+1
.
Proof. We first bound the height H(Pi ). Set m = n − hi + 1 and denote by y0 , . . . , ym the
columns of the Hankel matrix


y0
. . . ym
 ..
.. 
(6)
 .
. 
yhi −1
...
yn
which has rank hi by definition of hi (when hi = 1, this follows from xi = 0). Therefore the
vectors y0 , . . . , ym span over Q the whole space Qhi . Moreover m ≥ hi by Lemma 3. Now
we use the fundamental fact that these columns y0 , . . . , ym satisfy the linear recurrence
relations
(7)
(i)
(i)
a0 yj + · · · + ahi yj+hi = 0
5
(0 ≤ j ≤ m − hi )
M. Laurent
which are equivalent to (5). Then we can apply Lemma 1 which furnishes the upper bound
H(Pi ) Y 1/(m+1−hi ) = Y 1/(n−2hi +2)
where Y denotes the supremum of the absolute value of the maximal minors extracted
from the matrix (6). The estimation H(Pi ) Xiλ will be an immediate consequence of
the upper bound
(n−2hi +2)λ
Y Xi
(8)
.
In order to prove (8), let us write any maximal minor in the following way (setting
temporarily h = hi for simplicity):
yj1
yj2
...
yjh ..
..
..
.
.
.
.
.
.
..
..
..
y
y
.
.
.
y
j1 +h−1
j2 +h−1
jh +h−1
yj1
yj2 − yj1 θj2 −j1
...
yjh − yj1 θjh −j1
..
..
..
.
.
.
=
.
..
..
..
.
.
.
y
j2 −j1
jh −j1 y
−
y
θ
.
.
.
y
−
y
θ
j1 +h−1
j2 +h−1
j1 +h−1
jh +h−1
j1 +h−1
The entries in the first column of the right hand side matrix have absolute values ≤ Xi ,
while the other entries have absolute values Li . Using (2) we obtain the estimate
−(h −1)λ
Y Xi Lhi i −1 hi −1 Xi Xi+1 i
Observe now that
1 − (hi − 1)λ =
1−(hi −1)λ
Xi
n − 2hi + 3
n+1
Then (8) follows from the upper bound
n − 2hi + 3 ≤ 2(n − 2hi + 2)
since hi ≤ (n + 1)/2 by Lemma 3.
Now let us bound the absolute value of Pi (θ). Observe that
y0 Pi (θ) =
k=h
i
(i)
ak (y0 θk − yk )
k=0
by (5) with j = 0. It follows that
−λ
|Pi (θ)| Xi−1 H(Pi )Li Xi−1+λ Xi+1
by (2) and our bound for H(Pi ).
6
.
Simultaneous approximation of the successive powers of a real number
5. Proof of the Theorem.
The two following lemmas enable us to avoid the use of some suitable version of
Gel’fond’s criterion which was employed in [3].
We henceforth denote by (z0 , . . . , zn ) the coordinates of the vector xi−1 .
Lemma 5. Suppose that i is large enough. Then the coordinates of the vector xi−1 satisfy
the same linear recurrence relations
(i)
(i)
a0 zj + · · · + ahi zj+hi = 0
(9)
(0 ≤ j ≤ n − hi )
that the coordinates of the vector xi .
Proof. For j = 0, . . . , n − hi let us use the formula
zj Pi (θ) =
(i)
a0 zj
+ ··· +
(i)
ahi zj+hi
+
k=h
i
(i)
ak (zj θk − zj+k ).
k=0
It follows that
k=h
i
(i)
−λ
ak zj+k Xi−1 |Pi (θ)| + H(Pi )Li−1 Xi−1 Xiλ−1 Xi+1
+
k=0
by Lemma 4 and (2). The left hand side of the above inequality is an integer which vanishes
necessarily if 1 .
Let us view the equalities (9) as a non zero linear relation between the rows of the
Hankel matrix


z0 . . . zn−hi
 ..
.. 
 .
. 
zhi
...
zn
whose rank is therefore at most hi . It follows that hi−1 ≤ hi . Since the sequence of positive
integers (hi )i≥1 is bounded, we deduce that hi = h for large value of i, where h is some
positive integer independent of i.
Lemma 6. Suppose i is large enough. Then we have the equality
Pi = ±Pi−1 .
Proof. Now we consider the Hankel matrix

...
z0
 ..
 .
zh−1
...
7

zn−h+1

..

.
zn
M. Laurent
constructed with the coordinates of xi−1 . It has rank h and its columns z0 , . . . , zn−h+1
satisfy the linear relations
(i)
(i)
a0 zj + · · · + ah zj+h = 0
(10)
(0 ≤ j ≤ n − 2h + 1)
which are equivalent to (9). On the other hand, recall also the relations
(11)
(i−1)
a0
(i−1)
zj + · · · + ah
deduced from (7) with i replaced
 (i)
a0

 0
Ai = 
 .
 ..
0
zj+h = 0
(0 ≤ j ≤ n − 2h + 1)
by i − 1. Let
(i)
...
(i)
a1
..
a1
a0
..
.
...
(i)
.
0
ah
(i)
0
...
..
.
(i)
a0
ah
(i)
(i)
a1

... 0
.. 
..
.
. 


..
.
0 
(i)
. . . ah
be the (n − 2h + 2) × (n − h + 2) matrix formed with the coefficients of the linear relations
(10) which are linearly independent. The n − 2h + 2 rows of Ai span the Q-vector space of
linear relations connecting the n − h + 2 columns z0 , . . . , zn−h+1 , since these vectors span
the whole space Qh . Observe now that Ai−1 is the matrix associated to the linear relations
(11) between the same vectors. Therefore the rows of the matrices Ai and Ai−1 span the
same Q-vector space in Qn−h+2 . By Lemma 2, the matrices Ai and Ai−1 are proportional
and Pi = ρPi−1 for some ρ ∈ Q× . Since the coefficients of the polynomials Pi and Pi−1
are coprime integers, we have ρ = ±1.
We are now able to conclude the proof of our Theorem. By Lemma 6, we know that
for large i, Pi = ±P for some fixed non zero polynomial P with integral coefficients of
degree ≤ h ≤ (n + 1)/2. When i tends to infinity, Lemma 4 implies that P (θ) = 0. We
have thus proven that θ is an algebraic number of degree ≤ (n + 1)/2.
6. Deduction of the Corollary.
The link between simultaneous rational approximation and approximation by algebraic integers is achieved by the following result due to H. Davenport and W. M. Schmidt,
which we reproduce here in our setting. See Lemma 1 of [3], and [1] for the fact that the
degree of the algebraic approximations can be prescribed and not only bounded.
Lemma 7. Let n be an integer ≥ 2, let θ be a real number and let λ be a positive real
number which is not an exponent of simultaneous approximation to θ, . . . , θn−1 . Then
there are infinitely many algebraic integers α of degree n which satisfy
0 < |θ − α| H(α)−1−1/λ .
8
Simultaneous approximation of the successive powers of a real number
The Corollary follows immediately from Lemma 7, since the Theorem asserts that
λ=
1
(n − 1)/2
is not an exponent of simultaneous approximation to θ, . . . , θn−1 , provided that θ is not
an algebraic number of degree ≤ (n − 1)/2.
References
[1] Y. Bugeaud and O. Teulié : Approximation d’un nombre réel par des nombres
algébriques de degré donné, Acta Arith., 93-1 (2000), 77-86 ;
[2] J.W.S. Cassels : Introduction to Diophantine Approximation, Cambridge University
Press, 1965.
[3] H. Davenport and W. M. Schmidt : Approximation to real numbers by algebraic
integers, Acta Arith., 15 (1969), 393–416.
[4] A. Dress, N. Elkies and F. Lucas : A Characterization of Mahler’s generalized Liouville
numbers by simultaneous rational approximation, preprint 2001, http://www.math.
harvard.edu/∼ elkies/Um .tex
[5] D. Roy : Approximation to real numbers by cubic algebraic integers, preprint 2002.
[6] D. Roy and M. Waldschmidt : Diophantine approximation by conjugate algebraic
integers, preprint 2002, http://arXiv.org/abs/math.NT/0207102
Michel LAURENT
Institut de Mathématiques de Luminy, CNRS.
163 Avenue de Luminy, Case 907.
13288 MARSEILLE Cédex 9
FRANCE.
e-mail : [email protected]
9