Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

0.2 MB·

by  Johannes Schleischitz

RATIONAL APPROXIMATION TO ALGEBRAIC VARIETIES AND A
NEW EXPONENT OF SIMULTANEOUS APPROXIMATION
JOHANNES SCHLEISCHITZ
Abstract. This paper deals with two main topics related to Diophantine approxima-
6
1 tion. Firstly,weshowthatifa pointonanalgebraicvarietyis approximablebyrational
0 vectors to a sufficiently large degree,the approximatingvectors must lie in the topolog-
2 icalclosureofthe rationalpointsonthe variety. Inmanyinterestingcases,inparticular

r if the set of rational points on the variety is finite, this closure does not exceed the set
a
ofrationalpointsonthe varietyitself. Thisresultenableseasierproofsofseveralknown
M
results as special cases. The proof can be generalized in some way and encourages to
define a new exponent of simultaneous approximation. The second part of the paper is
8
devoted to the study of this exponent.

]
T
N
Supported by FWF grant P24828
.
h
Institute of Mathematics, Department of Integrative Biology, BOKU Wien, 1180, Vienna, Austria.
t
a [email protected]
m
[
Math subject classification: 11J13,11J82,11J83

2
Key words: exponents of Diophantine approximation, rational points on varieties, continued fractions
v
3
1
8 1. Introduction
2
0
. In this paper we study certain aspects concerning the simultaneous approximation of
1
0 vectors ζ ∈ Rk by rational vectors. In the classical setting of simultaneous approxima-
6
tion the approximating rational vectors are of the form (p /q,...,p /q) ∈ Qk and the
1 1 k
: maximum of |ζ − p /q| is compared with the size of (large) q. In Sections 1,2 we stick
v i i
i to this classical setting and derive a new result concerning very well approximable points
X
on varieties that generalizes several results that have been established. This main result
r
a has a natural extension to the case where the denominators of the rational approxima-
tions may differ. Motivated by this we will introduce a new exponent of simultaneous
approximation in Section 3 and study its properties.
We first introduce some notation.
Definition 1.1. Let k ≥ 1 be an integer. For a function ψ : R → R let H k ⊆ Rk be the
ψ
set of points ζ = (ζ ,...,ζ ) approximable to degree ψ, that is such that
1 k
max |xζ −y | ≤ ψ(x)
j j
1≤j≤k
has a solution (x,y ,...,y ) ∈ Zk+1 for arbitrarily large values of x. If ψ(x) = x−µ for
1 k
µ > 0, we will also write H k for H k and refer to ζ as approximable to degree µ.
µ ψ
1
2 JOHANNESSCHLEISCHITZ
Dirichlet’s Theorem can be formulated in the way that H k equals the entire space
1/k
Rk. Thus only functions ψ(x) ≤ x−1/k for large x resp. parameters µ > 1/k are of
interest. Furthermore it is known thanks to Khintchine [8] that the set H k for any
1/k+δ
fixedδ > 0hask-dimensionalLebesguemeasure0. Ontheotherhand, theset∪ H k
δ>0 1/k+δ
often referred to as (simultaneously) very well approximable vectors, has full Hausdorff
dimension k, see [7]. As usual denote by k.k the distance of a real number to the nearest
integer. Next we define constants closely related to the sets H k that have been intensely
µ
studied.
Definition 1.2. Let k ≥ 1 be an integer. For ζ = (ζ ,...,ζ ) ∈ Rk let ω (ζ) be the
1 k k
exponent of classical k-dimensional rational approximation, i.e. the supremum of ν > 0
such that
max kxζ k ≤ x−ν
j
1≤j≤k
has infinitely many integral solutions x. Similarly, let ω (ζ) be the supremum of µ such
k
that the system
0 < x ≤ X, max kxζ k ≤b X−µ
j
1≤j≤k
has an integral solutions x for every large parameter X.
The sets H k coincide with the sets {ζ ∈ Rk : ω (ζ) ≥ µ} for every µ > 0, respectively.
µ k
For the special case of ζ successive powers of a real number this leads to the quantities
λ ,λ defined by Bugeaud and Laurent [5].
k k
Definition 1.3. Let k ≥ 1. For ζ ∈ R define λ (ζ) as the supremum of real µ such that
b k
max kxζjk ≤ x−µ
1≤j≤k
has arbitrarily large solutions x. Similarly, let λ (ζ) be the supremum of µ such that the
k
system
b
0 < x ≤ X, max kxζjk ≤ X−µ
1≤j≤k
has an integral solutions x for every large parameter X.
In particular the classic one-dimensional approximation constants λ (ζ) for ζ ∈ R is
1
defined as the supremum of real µ such that kxζk ≤ x−µ has arbitrarily large solutions
x. For k = 1 obviously ω (ζ) = λ (ζ) and consequently the sets H 1 coincide with the
1 1 µ
set {ζ ∈ R : λ (ζ) ≥ µ}. Clearly 1/k ≤ λ (ζ) ≤ λ (ζ) for all k and ζ such as
1 k k
λ (ζ) ≥ λ (ζ) ≥ ··b· , λ (ζ) ≥ λ (ζ) ≥ ···
1 2 1 2
for every ζ. Moreover, we have λ (ζ) = 1 forbevery irbrational ζ and λ (ζ) = 1/k for
1 k
almost all ζ in the sense of Lebesgue measure [17]. For further results concerning the
b
spectrum of the exponents see for example [3], [5], [14].
Finally we introduce the absolute degree of a polynomial.
Definition 1.4. For a monomial M := aXj1···Xjk with a ∈ Q\{0} let j + ···+ j
1 k 1 k
be the total degree of M. For P ∈ Q[X ,...,X ] define the absolute degree of P as the
1 k
maximum of the total degrees of the monomials involved in P.
RATIONAL APPROXIMATION TO ALGEBRAIC VARIETIES AND A NEW EXPONENT OF SIMULTANEOUS APPROXIMA
2. A result on approximation to varieties
Theorem 2.1 is the main result of this section. Its proof is not difficult and based onthe
fact that if a polynomial with rational coefficients of absolute degree r does not vanish
at some point (y /x,...,y /x) then the evaluation is bounded below essentially by x−r.
1 k
We partly state it because in view of Theorem 3.5 below it will help to motivate the new
exponent we will introduce in Section 3.
Theorem 2.1. Let P ∈ Q[X ,...,X ] of absolute degree r and V be the variety defined
1 k
by
V = {(X ,X ,...,X ) ∈ Rk : P(X ,X ,...,X ) = 0}.
1 2 k 1 2 k
Denote T := V ∩Qk the rational points on V. Let ψ : R → R be any function with the
property ψ(t) = o(t−r+1) as t → ∞. Then T ⊆ H k ∩ V ⊆ T , where T denotes the
ψ
topological closure of T with respect to the usual Euclidean metric.
Proof. Clearly we may assume P ∈ Z[X ,...,X ]. It also obvious that T ⊆ H k ∩
1 k ψ
V for an arbitrary function ψ, since given (p /q,...,p /q) ∈ T it suffices to take
1 k
(x,y ,...,y ) = (Mq,Mp ,...,Mp )theintegralmultiplesofthevector(M ∈ {1,2,...})
1 k 1 k
in Definition 1.1. We must prove that H k ∩V ⊆ T for ψ(t) = o(t−r+1).
ψ
Let ζ = (ζ ,...,ζ ) ∈ V \T . We have to show ζ ∈/ H k. Assume ζ ∈ H k. By
1 k ψ ψ
definition we have
y
ζ − j ≤ ψ(x)x−1, 1 ≤ j ≤ k
j
x
(cid:12) (cid:12)
for arbitrarily large x. H(cid:12)ence we(cid:12) can write ζ = y /x + ǫ with |ǫ | ≤ ψ(x)x−1 for
(cid:12) (cid:12) j j j j
1 ≤ j ≤ k. Since ζ ∈/ T , there exists some open neighborhood U ∋ x of x such that
U∩T = ∅, or in other words there is no rational point in U∩V. Observe that P is C∞ on
Rk, thus inU thepartial derivatives P ,...,P areuniformly boundedby some constant
x1 xk
C in absolute value. We may assume x to be large enough that (y /x,...,y /x) ∈ U.
1 k
With repeated use of (one-dimensional) Taylor Theorem parallel to the coordinate axes
we obtain
y y y y
1 k 1 k
(1) 0 = P(ζ) = P +ǫ ,..., +ǫ = P ,..., +ǫ P (t )+···+ǫ P (t )
x 1 x k x x 1 x1 1 k xk k
(cid:16) (cid:17) (cid:16) (cid:17)
where t ∈ U. Thus
j
y y
(2) P(ζ)−P 1,..., k ≤ kC ·max|ǫ | ≤ kC ·ψ(x)x−1.
j
x x
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
Since (y1/x,...,(cid:12)yk/x) ∈ V ∩U which ha(cid:12)s empty intersection with Qk we derive
P(y /x,...,y /x) 6= 0.
1 k
Thus and since P ∈ Z[X ,...,X ] has absolute degree r we obtain |P(y /x,...,y /x)| ≥
1 k 1 k
x−r. Hence and since ψ(t) = o(t−r+1), for large x from (1) and (2) we infer
y y y y 1
|P(ζ)| ≥ P 1,..., k − P(ζ)−P 1,..., k ≥ x−r −kC ·x−1ψ(x) ≥ x−r.
x x x x 2
(cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12)
This contra(cid:12)(cid:12)dicts P(ζ) = 0.(cid:12)(cid:12)Hen(cid:12)(cid:12)ce indeed ζ ∈/ H k and(cid:12)(cid:12)the proof is finished. (cid:3)
ψ
The theorem in particular applies if T is finite.
4 JOHANNESSCHLEISCHITZ
Corollary 2.2. With the definitions and assumptions of Theorem 2.1 assume that the
set T of rational points on V is finite. Then H k ∩V = T .
ψ
Corollary 2.2 contains various known results as special cases. For example the Fermat
curve defined as the set of zeros of P(X,Y) = Xk +Yk −1 has only possibly the trivial
points {(±1,0),(0,±1)} approximable to degree greater k−1, which was established by
Bernik and Dodson [1, p. 94]. Corollary 2.2 also implies one of the two claims of the
main result of [6, Theorem 1.1] by Dru¸tu. Concretely it asserts that for a quadratic form
Q in arbitrary many variables, if there are no rational points on the variety defined by
Q(X) −1 = 0, then there are no points on this variety approximable to degree greater
than 1. In fact Theorem 2.1 generalizes [6, Lemma 4.1.1] which readily implied this claim.
However, it should be pointed out that the main and much more technical result of [6,
Theorem 1.1] is the other claim, which provides a formula for the Hausdorff dimension
for the variety as above in the case that it contains rational points. Observe also that
Corollary 2.2 implies that an elliptic curve of rank 0 contains only finitely many points
approximable to degree larger than 3 by rational vectors. We want to add that a very
similar result was proved for very well approximable points on surfaces parametrized by
polynomials with rational coefficients, see [2, Lemma 1].
The case that T in Theorem 2.1 is infinite but consists solely of isolated rational points
that may have some non-rational limit point on V (observe V is closed) is of interest.
The question arises how large the set T \T of such limit points can be, for example in
sense of Hausdorff measure. It is already not obvious how to find an algebraic variety
where T is infinite and consists solely of isolated points.
3. A new exponent of simultaneous approximation
The proof of Theorem 2.1 can be extended in some way to a similar Diophantine
approximation problem that seems so far unstudied in the literature. We first define the
new exponent of simultaneous approximation below and derive some propoerties, andwill
return to the connection with Section 2 in Theorem 3.5.
For a real function ψ(t) that tends to 0 as t → ∞ let Zk be the set of ζ = (ζ ,...,ζ ) ∈
ψ 1 k
Rk such that the system
(3) 0 < min |x | ≤ max |x | ≤ X, max kx ζ k ≤ ψ(X)
j j j j
1≤j≤k 1≤j≤k 1≤j≤k
has a solution (x ,...,x ) ∈ Zk for arbitrarily large X. Moreover write Zk instead of
1 k ν
Zk when ψ(t) = t−ν with a parameter ν > 0. Further denote by χ (ζ) the supremum of
ψ k
exponents ν for which ζ ∈ Zk, such that
ν
Zk = {ζ ∈ Rk : χ (ζ) ≥ ν}.
ν k
Obviously Zk ⊇ H k for all k ≥ 1 and any ζ ∈ Rk for any function ψ, with equality if
ψ ψ
k = 1. In particular χ (ζ) ≥ ω (ζ) for all k ≥ 1 and all ζ ∈ Rk. Moreover Zk = Rk by
k k 1
the uniform version of Dirichlet’s Theorem applied to any single ζ . Furthermore the k-
j
dimensional exponent is trivially bounded above by the minimum of the one-dimensional
constants λ (ζ ). As stated in Section 1 each of these single exponents equals 1 also for
1 j
RATIONAL APPROXIMATION TO ALGEBRAIC VARIETIES AND A NEW EXPONENT OF SIMULTANEOUS APPROXIMA
almost all ζ ∈ R in terms of Lebesgue measure. Hence for almost all ζ ∈ Rk we have
χ (ζ) = 1. Moreover by Roth’s Theorem χ (ζ) = 1 if there is at least one irrational
k k
algebraic element among the ζ .
j
We can reformulate the above observations by the formula
(4) max{1,ω (ζ)} ≤ χ (ζ) ≤ min λ (ζ ).
k k 1 j
1≤j≤k
Recall the one-dimensional constants λ (ζ) are determined by the continued fraction
1
expansion of ζ. Roughly speaking, the exponent χ somehow measures the distances of
k
denominators of those convergents p /q , which lead to very good approximation |p /q −
j j j j
ζ | of the ζ , compared to the single q . The situation is different for the exponents ω ,
j j j k
where denominators of continued fractions of single ζ lead to a large exponent ω only
j k
if their lowest common multiple is small compared to the smallest single q . Roughly
j
speaking the exponents χ measure something in between the separate one-dimensional
k
best approximations λ of the single ζ and the classical simultaneous approximation
1 j
constants ω . Another relation between χ and ω is given by the following easy lemma
k k k
where this phenomenon becomes apparent.
Lemma 3.1. Let k ≥ 1 and ζ ∈ Rk. We have
χ (ζ)−k +1
k
ω (ζ) ≥ .
k
k
Proof. Assume the system
0 < max |q | ≤ Q, max kq ζ k ≤ Q−ν,
j j j
1≤j≤k 1≤j≤k
is satisfied. Then 0 < q ···q ≤ Qk and
1 k
kq q ···q ζ k ≤ (q q ···q q ···q )kq ζ k ≤ Qk−1−ν = (Qk)−(ν−k+1)/k, 1 ≤ j ≤ k.
1 2 k j 1 2 j−1 j+1 k j j
The claim follows since we may let ν arbitrarily close to χ (ζ). (cid:3)
k
Uniform exponents can be defined similarly to the classical simultaneous Diophantine
approximation constants, but since Dirichlet’s Theorem is uniform in the parameter Q
again (for irrational ζ )
j
1 = max{1,ω (ζ)} ≤ χ (ζ) ≤ min λ (ζ ) = 1,
k k 1 j
1≤j≤k
and hence b b b
χ (ζ) = 1
k
for all ζ ∈/ Qk (for ζ ∈ Q we have λ (ζ ) = ∞). We formulate some questions concern-
1 jb
ing the constants χ similar to well-known (partially answered) problems for the classic
k b
exponents ω ,λ , see for example [3, Problem 1-3]. By the spectrum of χ we will mean
k k k
the set {χ (ζ) : ζ ∈ T } ⊆ R of values taken by χ in the set T ⊆ Rk of ζ ∈ Rk which
k k k k
are linearly independent together with {1} over Q.
Problem 3.2. Is the spectrum of χ equal to [1,∞]? Find explicit constructions of
k
ζ ∈ Rk with prescribed values of χ (ζ).
k
6 JOHANNESSCHLEISCHITZ
Problem 3.3. Metric theory: For λ ∈ [1,∞] determine the Hausdorff dimensions of the
sets
dim({ζ ∈ Rk : χ (ζ) = λ}), dim({ζ ∈ Rk : χ (ζ) ≥ λ}).
k k
Problem 3.4. What about Problems 3.2, 3.3 for the restriction of ζ to certain manifolds
in Rk? In particular the Veronese curve which consists of the vectors ζ = (ζ,ζ2,...,ζk)
for ζ ∈ R.
Concerning Problem 3.3, we point out that the estimates
k +1 k +1 k(k +1)
(5) ≤ dim({ζ ∈ Rk : χ (ζ) ≥ λ}) ≤ =
1+λ k 1+ λ−k+1 1+λ
k
hold, where the right inequality is non-trivial only for λ > k. Indeed Jarn´ık [7] proved
k +1
= dim({ζ ∈ Rk : λ (ζ) ≥ λ}) = dim({ζ ∈ Rk : λ (ζ) = λ})
k k
1+λ
for λ ∈ [1/k,∞], which in combination with χ (ζ) ≥ ω (ζ) and Lemma 3.1 respectively
k k
proves the inequalities in (5) respectively.
Concerning Problem 3.4 for varieties, a slight modification of the proof of Theorem 2.1
shows the following.
Theorem 3.5. Let P ∈ Q[X ,...,X ] of absolute degree r and V be the variety defined
1 k
by
V = {(X ,X ,...,X ) ∈ Rk : P(X ,X ,...,X ) = 0}.
1 2 k 1 2 k
Denote T := V ∩Qk the rational points on V. Let ψ : R → R be any function with the
property ψ(X) = o(X−kr+1) as X → ∞. Then T ⊆ Zk ∩V ⊆ T .
ψ
Proof of Theorem 3.5. Proceed precisely as in the proof of Theorem 2.1, and notice that
for general fractions z := (p /q ,...,p /q ) we still have the lower bound |P(z)| ≥
1 1 k k
q−rq−r···q−r ≥ Q−kr. (cid:3)
1 2 k
Remark 3.6. The proof shows that forthe largeclass of varieties the exponent kr−1 can
be readily improved. This is the case if the polynomial does not contain all monomials
a Xr,a Xr,··· ,a Xr with non-zero coefficients a 6= 0. More precisely the condition
1 1 2 2 k k i
ψ(x) = o(x−r+1), with r := k r ≤ kr where r ≤ r is the degree of P(X ,...,X )
j=1 j j 1 k
in the variable Xj, suffices toPobtain the result of Theorem 3.5. In particular if P is of
the form P(X ,...,X ) = Xr1Xr2···Xrk − l /l for l /l ∈ Q, then ψ(x) = o(x−r+1)
1 k 1 2 k 1 2 1 2
is sufficient. More generally this applies for P(X ,...,X ) = (p/q)Xr1Xr2···Xrk +
1 k 1 2 k
Q(X ,...,X ) for p/q ∈ Q and any Q ∈ Q[X ,...,X ] of degree at most r in the
1 k 1 k j
variable X for 1 ≤ j ≤ k.
j
Wewanttopointoutsomeconsequences andinterpretationsofTheorem3.5, whichalso
aim to shed more light on the meaning of the exponent χ in general. Recall a Liouville
k
number is an irrational real (and thus transcendental by Liouville’s Theorem) number
that satisfies λ (ζ) = ∞. It is shown in [9] that for any countable set of continuous
1
strictly monotonic functions f : A → B with A,B subsets of R, there are uncount-
i
ably many Liouville numbers ζ ∈ A such that f (ζ) is again a Liouville number for all
i
RATIONAL APPROXIMATION TO ALGEBRAIC VARIETIES AND A NEW EXPONENT OF SIMULTANEOUS APPROXIMA
i. See also [13], [16]. Let C be any curve in Rk for arbitrary k defined by algebraic
equations. Then C can be almost everywhere locally parametrized by such functions
f = id,f ,...,f , in other words any (ζ ,...,ζ ) ∈ C can be written ζ = f (ζ) for
0 1 k−1 1 k i+1 i
0 ≤ i ≤ k−1. Hence there are uncountably many Liouville points on the curve, by which
we mean that every coordinate is a Liouville number. On the other hand, if C is a variety
that contains no rational point, by Theorem 3.5 there are also no points simultaneously
approximable to asufficiently largefinitedegree inthesense of largeχ (ofcourse also not
k
for ω ). This emphasizes that on algebraic curves there is a huge difference between the
k
minimum of the one-dimensional classical constants λ (ζ ) and the constants χ (ζ). For
1 j k
0 ≤ i ≤ k−1denote by (p /q ) the sequence ofconvergents off (ζ). Then theabove
n,i n,i n≥1 i
result means that for the Liouville numbers ζ,f (ζ),...,f (ζ) in the parametrization
1 k−1
there do not exist infinitely many convergents p /q ,...,p /q whose denomina-
.,0 .,0 .,k−1 .,k−1
tors q ,0 ≤ i ≤ k − 1 are all of ”similar” largeness. The analogue phenomenon holds
.,i
for all algebraic surfaces of dimension larger one as well. Indeed, if the dimension of the
variety is locally k, then we can write the variety locally as (ζ ,...,ζ ,ψ (ζ),...,ψ (ζ))
1 k 1 r
with ζ = (ζ ,...,ζ ) and C∞ functions ψ in some open U subset of Rk. We fix the first
1 k j
k−1 coordinates as Liouville numbers in some open subset of Rk−1 (i.e. we pick Liouville
numbers in the open projection set V ⊆ U of U to the first k − 1 coordinates) and the
analogue result follows from the one-dimensional case.
Concerning thespectrum ofthequantities χ (ζ)thenext theoremisrathersatisfactory.
k
Theorem 3.7. Let k ≥ 2 an integer and λ ,λ ,...,λ ,w real numbers that satisfy 1 ≤
1 2 k
w ≤ min λ . Then there exist uncountably many vectors (ζ ,ζ ,...,ζ ) ∈ Rk that
1≤j≤k j 1 2 k
are Q-linearly independent together with {1} and such that λ (ζ ) = λ for 1 ≤ j ≤ k and
1 j j
χ (ζ ,...,ζ ) = w.
k 1 k
The condition w ≤ min λ is necessary in view of (4). It would be nice to have
1≤j≤k j
some additional relation between χ and ω included. In Theorem 3.9, which treats the
k k
special case of the Veronese curve, a connection to the constants λ will be given provided
k
the parameter is at least 2. We emphasize that Theorem 3.7 answers Problem 3.2.
Corollary 3.8. The spectrum of χ equals [1,∞].
k
Now we turn towards Question 3.4. We restrict to ζ on the Veronese curve and denote
the exponent χ (ζ) = χ(ζ,ζ2,...,ζk). Since χ (ζ) ≥ λ (ζ), from [3, Lemma 1] we infer
k k k
λ (ζ)−k +1
1
(6) χ (ζ) ≥ .
k
k
For large parameters λ (ζ) and special choices of ζ, very similarly constructed as in the
1
proof of [3, Theorem 1] by Bugeaud, we will show in Theorem 3.9 that there is equality
in (6). The proof of this is among other things based on the fact that there cannot be
two good approximations p/q,p′/q′ to ζ with q,q′ that do not differ much. Some parts of
the proof also involve similar ideas as the proof of [10, Theorem 6.2] or [15, Lemma 4.10].
Our main result concerning Question 3.4 is the following.
8 JOHANNESSCHLEISCHITZ
Theorem 3.9. Let k ≥ 1 be an integer. For λ ∈ [2,∞] real transcendental ζ can be
explicitly constructed such that χ (ζ) = λ (ζ) = λ. In particular, the spectrum of χ on
k k k
the Veronese curve contains [2,∞].
See also the remarks subsequent to the proof of Theorem 3.9 that relate Theorem 3.9
and ζ constructed in the proof with classical approximation constants. We end by stating
the natural conjecture.
Conjecture 3.10. The spectrum of χ on the Veronese curve equals [1,∞].
k
4. Proofs of Theorem 3.7 and Theorem 3.9
Theproofsheavilyusethetheoryofcontinuedfractions. Anyirrationalrealnumberhas
a unique representation as ζ = a +1/(a +1/(a +···)) for positive integers a that can
0 1 2 j
be recursively determined. This is called the the continued fraction expansion of ζ and we
also write ζ = [a ;a ,a ,...]. The evaluation of any finite subword r /s = [a ;a ,...,a ]
0 1 2 l l 0 1 l
is called convergent to ζ and satisfies |r /s −ζ| ≤ s−2. More precisely we have
l l l
a r 1
l+2 l
(7) ≤ −ζ ≤ .
s s (cid:12)s (cid:12) s s
l l+2 (cid:12) l (cid:12) l l+1
(cid:12) (cid:12)
Recall also the inductive formulas rl+1 =(cid:12)al+1rl+(cid:12) rl−1,sl+1 = al+1sl+sl−1. We will utilize
also the following well-known result.
Theorem 4.1 (Legendre). If for irrational ζ the inequality
1
|qζ −p| ≤ q−1
2
has an integral solution (p,q) ∈ Z2 then p/q is a convergent of ζ in the continued fraction
expansion.
Proof of Theorem 3.7. First we do not take care of the Q-linear independence condition
and in the end describe how to modify the constructions below to ensure this additional
condition. Without loss of generality 1 ≤ λ ≤ λ ≤ ··· ≤ λ . Let
1 2 k
ζ = [0;1,1,...,1,h ,1,1...,1,h ,1,...]
j j,1 j,2
for the positions at which the h 6= 1 are such as the values h to be determined
j,i j,i
later. For i ≥ 1 denote r /s the convergent [1,...,1,h ]. Observe that by elementary
j,i j,i j,i
estimates for continued fractions related to (7), for any convergent r/s not equal to some
r /s we have |sζ −r| ≥ (1/3)s−1. Hence and by Theorem 4.1, for w > 1, every large
j,i j,i j
solution of the system (3) for ψ(t) = t−(1+w)/2 has each x an integral multiple of some
j
s . Similarly, if w = 1, the argument applies with ψ(t) = t−1−ǫ for every ǫ > 0. Hence
j,i
we may restrict x of the form s .
j j,i
First define h with sufficiently large differences h −h recursively in a way that
1,i 1,i+1 1,i
log|ζ s −r |
1 1,i 1,i
lim − = λ .
1
i→∞ logs1,i
RATIONAL APPROXIMATION TO ALGEBRAIC VARIETIES AND A NEW EXPONENT OF SIMULTANEOUS APPROXIMA
This is clearly possible and leads to ζ = lim r /s that satsifies λ (ζ ) = λ . Now
1 i→∞ 1,i 1,i 1 1 1
we choose h of the remaining ζ ,...,ζ with the properties
j,i 2 k
log|ζ s −r |
1 j,i j,i
(8) lim − = λ ,
j
i→∞ logsj,i
and
logs w
j,i
(9) lim = .
i→∞ logs1,i λj
Such a choice is again possible. To satisfy (9) we just have to stop reading ones in the
continued fraction expansion at the right position, which is possible since by reading only
ones two successive denominators of convergents differ by a factor at most 2. Then to
guarantee (8) we just have to take the next partial quotient, that is some h , of the right
j,i
order.
We prove that the implied ζ have the desired properties. Observe that since w ≤ λ ≤
j 1
... ≤ λ and the gap between s and s can be arbitrarily large, we may assume
k 1,i 1,i+1
(10) s > s > s ··· > s , s > sλ1.
1,i 2,i 3,i k,i k,i+1 1,i
For X = s and q = s for 1 ≤ j ≤ k we have by construction
1,i j j,i
log|ζ s −r | log|ζ s −r |logs w
1 j,i j,i 1 j,i j,i j,i
lim − = lim − = λ = w.
j
i→∞ logX i→∞ logsj,i logX λj
Hence χ (ζ ,...,ζ ) ≥ w by the definition of the constant χ . On the other hand, we
k 1 k k
carried out above that we have to take each x = s for some i. Thus the optimal choices
j j,i
are given by X = s for some j. But (10) implies j = 1 since otherwise if X = s for
j,i j,i
j 6= 1 then s > X but
1,i
log|ζ s −r |
1 1,i−1 1,i−1
lim − < 1.
i→∞ logX
This would imply χ (ζ ,...,ζ ) = 1. In case of w > 1 this indeed gives a contradiction. It
k 1 k
follows in fact the choices carried out are optimal and thus χ (ζ ,...,ζ ) ≤ w, such that
k 1 k
thereisequality. Finally, inthecasew = 1theaboveconstructionimpliesχ (ζ ,...,ζ ) =
k 1 k
1 very similarly.
Finally we carry out how to guarantee that the vector ζ can be chosen Q-linearly
independent together with {1}, by a slight modification of the above construction. In
the process we can recursively choose ζ for 1 ≤ j ≤ k in turn not in the Q-span of
j
{1,ζ ,...,ζ }. First observe that ζ must be transcendental if λ (ζ ) > 1 by Roth
1 j−1 1 1 1
Theorem, and otherwise the claim of the theorem is a trivial consequence of (4) for any
Q-linearly independent vector ζ with first coordinate ζ anyway. For the recursive step
1
notethatthespanofj−1numbersiscountablebutwehaveatinfinitelymanypositionsat
least two choices of positions where to put h (it follows from the proof that the positions
j,i
are not completely determined but there is some freedom). Pigeon hole principle implies
there must be uncountably many choices for ζ and repeating this argument we obtain
j
uncountably many vectors that have Q-linearly independent coordinates. (cid:3)
10 JOHANNESSCHLEISCHITZ
Now we turn towards the proof of Theorem 3.9. It needs some preperation. First recall
Minkowski’s second lattice point Theorem [11] asserts that for a lattice Λ in Rk with
determinant detΛ and a central-convex body K ⊆ Rn of n-dimensional volume vol(K),
the product of the successive minima t ,...,t of K relative to Λ are bounded by
1 n
2k detΛ detΛ
≤ t t ···t ≤ 2k .
1 2 k
k! vol(K) vol(K)
Applied in dimension 2 and for the lattice Λ := {x+ζy : x,y ∈ Z} and the 0-symmetric
convex body K := {−Q ≤ x ≤ Q,−1/(2Q) ≤ y ≤ 1/(2Q)} it yields the following.
Q
Theorem 4.2 (Minkowski). Let ζ be a real number. Then for any parameter Q > 1 the
system
1
(11) |q| ≤ Q, |ζq−p| ≤
2Q
cannot have two linearly independent integral solution pairs (p,q).
Moreover, we need some facts on continued fractions which can be found in [12].
Theorem 4.3. For irrational ζ and every convergent p/q of ζ in lowest terms we have
|qζ −p| ≤ q−1.
More generally, for any parameter Q > 1 the system
1 ≤ q ≤ Q, |qζ −p| ≤ Q−1
has a solution (p,q) with p/q a convergent of ζ.
Callq ∈ Nabest approximationofζ ifkqζk = min1≤q′≤qkq′ζk. Asq → ∞thisinducesa
sequence of best approximations (that uniquely determines ζ). The following connection
to the continued fraction expansion of ζ is well-known.
Lemma 4.4 (Lagrange). The sequence of best approximations is induced by the sequence
of convergents to ζ. More precisely, the j-th element of the sequence is the denominator
of the j-th convergent to ζ.
The next Proposition is in fact also well-known. However, we give a proof based on
Theorem 4.3, Theorem 4.1 and the fact that for ζ = [a ;a ,...] with convergents r /s
0 1 n n
we have s = a s +s (where formally s = 1,s = 0). Observe by Lemma 4.4
n+1 n+1 n n−1 −2 −1
we have s = q for q the n-th best approximation.
n n n
Proposition 4.5. Let q ,q ,... be the sequence of best approximations of ζ = [a ;a ,···].
1 2 0 1
Let
logkq ζk logq log(a q )
n n+1 n+1 n
ν := − , η := , τ := .
n n n
logq logq logq
n n n
Then η −ν = o(1) and η −τ = o(1) as n → ∞.
n n n n