Beresnevich, V.

Bernik, V.

English

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Publ. Math. Debrecen, 56 (2000), no. 3-4, 263-269.
A. Baker’s conjecture and Hausdorff dimension
by
V. Beresnevich1 and V. Bernik2
(Dedicated to the 60th birthday of Professor Ka´lma´n Gyo¨ry)
Abstract. It this paper we discuss an application of the Hausdorff dimen-
sion to the set of very well multiplicatively approximable points (x, . . . , xn).
In 1998 D.Kleinbock and G.Margulis proved A.Baker’s conjecture stating that
this set is of measure zero. We show that for any natural n multiplicatively
approximable points (x, . . . , xn) to order 1 + ε form a set of Hausdorff dimen-
sion at least 2/(1+ ε). It is conjectured that this number is the exact value of
the dimension. We also prove this conjecture for n = 2.
Introduction. We will use the following notation. The Vinogradov sym-
bol ¿ (À) means ‘≤ (≥) up to a positive constant multiplier’; a ³ b is
equivalent to a¿ b¿ a. The Lebesgue measure of A ⊂ R is denoted by |A|.
We denote by Pn the set of polynomials P ∈ Z[x] with degP ≤ n. Given a
polynomial P (x) = anx
n + . . .+ a1x+ a0 ∈ Z[x], we define the height of P as
H(P ) = max{|a0|, . . . , |an|}.
Let ε > 0, n ∈ N and Sn(ε) denote the set of x ∈ R such that the inequality
|P (x)| < H(P )−n(1+ε) (1)
has infinitely many solutions P ∈ Pn. In 1932 K. Mahler, in his classification
of real numbers, conjectured that for any ε > 0 the Lebesgue measure of Sn(ε)
is zero. Mahler’s problem was settled by V. Sprindzuk [5] in 1964. The concept
of Hausdorff dimension (see [4]) makes it possible to differ sets of measure zero.
In particular, this was applied to Sn(ε). In 1970 A. Baker and W. Schmidt
[2] established a lower bound for dimSn(ε), the Hausdorff dimension of Sn(ε).
Later it was proved by V. Bernik [3] that this value is also an upper bound for
dimSn(ε) resulting in
dimSn(ε) =
n+ 1
n+ 1 + nε
. (2)
1beresnevich@im.bas-net.by Institute of mathematics, the Belarus Academy of Sciences
2bernik@im.bas-net.by Institute of mathematics, the Belarus Academy of Sciences
1
In 1975 A. Baker raised a problem by replacing the right hand side of (1)
with the function Π+(P )
−1−ε, where P (x) = anxn + . . . + a1x + a0 ∈ Pn and
Π+(P ) =
∏n
i=1max(1, |ai|). Given ε > 0 and n ∈ N, let Mn(ε) be the set of
x ∈ R such that the inequality
|P (x)| < Π+(P )−1−ε (3)
has infinitely many solutions P ∈ Pn. A. Baker [1] conjectured that for any
n ∈ N one has |Mn(ε)| = 0 for any ε > 0.
Notice that Baker’s conjecture is stronger than that of Mahler. Indeed,
since H(P )n ≥ Π+(P ), we have H(P )−(1+ε)n ≤ Π+(P )−(1+ε). Therefore, if (1)
is soluble infinitely often, then so is (3). In particular, it means that
Sn(ε) ⊂Mn(ε). (4)
Baker’s conjecture was proved by D. Kleinbock and G. Margulis [7] in 1998.
As in the case of Sn(ε), it is also of interest to determine the Hausdorff
dimension of Mn(ε). We will use the following properties [4]:
1) dimA ≤ dimB for any A,B ⊂ R with A ⊂ B;
2) dimA = supi=1,2,... dimAi, where A =
⋃∞
i=1Ai and Ai ⊂ R.
Conjectures and results. First of all, notice that
Mk(ε) ⊂Mn(ε) for any k, n ∈ N with k < n. (5)
It follows from (4) and (5) that S1(ε) ⊂ Mn(ε) for any n ∈ N. Therefore, we
have dimMn(ε) ≥ dimS1(ε). Now applying (2) gives
Theorem 1. For any n ∈ N and any ε > 0
dimMn(ε) ≥ 2
2 + ε
. (6)
Conjecture H1. For any n ∈ N and ε > 0 one has
dimMn(ε) =
2
2 + ε
.
This conjecture is trivial for n = 1. Indeed, it is easy to notice that for
any δ > 0 we have the inclusion M1(ε) ⊂ S1(ε− δ). Therefore, for any δ with
0 < δ < ε we have dimM1(ε) ≤ dimS1(ε − δ). By (2), we conclude that
2
dimM1(ε) ≤ 2/(2 + ε− δ). Since δ ∈ (0, ε) is arbitrary, we have dimM1(ε) ≤
2/(2 + ε). In this paper we also prove the conjecture for n = 2.
Theorem 2. For any ε > 0 we have
dimM2(ε) =
2
2 + ε
.
Proof of Theorem 2. By (6), it is sufficient to show that dimM2(ε) ≤
2/(2 + ε). Let {Ik}∞k=1 be a collection of closed intervals such that R \ {0} =⋃∞
k=1 Ik. The existence of such a collection is easily verified. Then, M2(ε) =
{0}⋃(⋃∞k=1M2(ε)⋂ Ik). Since dim{0} = 0, by property 2 of Hausdorff dimen-
sion above, we have dimM2(ε) ≤ supk=1,2,... dim(M2(ε)
⋂
Ik). Therefore, it is
sufficient to show that dim(M2(ε)
⋂
Ik) ≤ 2/(2 + ε) for any k. Let I be one of
the intervals Ik. There is no loss of generality in assuming that I = [a, b] with
0 < a < b <∞.
Let x ∈ I and P (t) = a2t2 + a1t + a0 ∈ P2 be a solution of (3). It follows
from (3) that
|a0| = |P (x)− a2x2 − a1x| ≤ Π+(P )−1−ε + |a2|x2 + |a1|x ≤
1 + |a2|b2 + |a1|b ≤ (1 + b+ b2)max{|a1|, |a2|}.
Therefore, we have
max{|a1|, |a2|} ≤ H(P ) ≤ (1 + b+ b2)max{|a1|, |a2|}. (7)
Now define the constant
C = min(a, 1/2)/(1 + b+ b2). (8)
Let M12 (ε, I) be the subset of M2(ε)∩ I consisting of x ∈ I such that there are
infinitely many P ∈ P2 satisfying{ |P (x)| < Π+(P )−1−ε,
|P ′(x)| < CH(P ). (9)
Let x ∈ I and P (t) = a2t2+a1t+a0 be a solution of (9). We have the following
two possibilities:
1) |a2| ≥ |a1|;
2) |a1| ≥ |a2|.
3
Consider the first one. It follows from (7) and (9) that
|2x+ a1/a2| ≤ CH(P )/|a2| ≤ C(1 + b+ b2) ≤ a.
Since x ≥ a, we have |a1/a2| = |2x − (2x + a1/a2)| ≥ |2x| − |2x + a1/a2| ≥
2a− a = a. Therefore, we obtain a ≤ |a1/a2| ≤ 1.
Consider the other possibility: |a1| ≥ |a2|. It follows from (7) and (9) that
|2xa2/a1 + 1| ≤ CH(P )/|a1| ≤ C(1 + b+ b2) ≤ 1/2.
Hence, |2xa2/a1| = |1−(2xa2/a1+1)| ≥ 1−|2xa2/a1+1| ≥ 1−1/2 = 1/2. Since
x ≤ b, we have |a2/a1| ≥ 1/(4b). Therefore, we obtain 1/(4b) ≤ |a2/a1| ≤ 1.
As a result we conclude that |a1| ³ |a2| for both the possibilities. Moreover,
by (7), we have |a1| ³ |a2| ³ H(P ). Therefore, P+(P ) ³ H(P )2 and the first
inequality of (9) implies that
|P (x)| ¿ H(P )−2(1+ε). (10)
Now if x ∈M12 (ε, I), then inequality (10) holds for infinitely many polynomials
P ∈ P2 and for any δ > 0 the inequality |P (x)| < H(P )−2(1+ε−δ) has infinitely
many solutions P ∈ P2. It follows that M12 (ε, I) ⊂ S2(ε − δ) for any δ with
0 < δ < ε. By (2), we obtain
dimM12 (ε, I) ≤ dimS2(ε− δ) =
3
3 + 2(ε− δ) .
Since δ ∈ (0, ε) is arbitrary, we get
dimM12 (ε, I) ≤
3
3 + 2ε
<
2
2 + ε
. (11)
Now we consider the set M22 (ε, I) = (M2(ε) ∩ I) \M12 (ε, I). It is easy to
verify that for any x ∈M22 (ε, I) the system{ |P (x)| < Π+(P )−1−ε,
|P ′(x)| ≥ CH(P ) (12)
holds for infinitely many polynomials P ∈ P2. Given a polynomial P ∈ P2, let
σ(P ) denote the set of x ∈ I satisfying (12). It is easy to notice that σ(P ) is a
union of at most three intervals, say σi(P ) with i = 1, 2, 3. Also if x ∈M22 (ε, I)
then x belongs to σi(P ) for infinitely many different polynomials P ∈ P2.
4
Fix P ∈ P2 and x, y ∈ σi(P ). By the Mean Value Theorem, we have
P (x)− P (y) = P ′(θ)(x− y), where θ is a point between x and y. Since σi(P )
is an interval, θ ∈ σi(P ) and, therefore, |P ′(θ)| ≥ CH(P ). Hence,
|x− y| ≤ |P (x)|+ |P (y)||P ′(θ)| ≤
2Π+(P )
−1−ε
CH(P )
.
Thus,
|σi(P )| ¿ Π+(P )−1−ε ·H(P )−1. (13)
Let 2/(2 + ε) < ρ < 1. We have the following inequality
∑
P∈P2
3∑
i=1
|σi(P )|ρ ¿
∞∑
k=0
∞∑
l=0
∑
2k≤|a1|<2k+1
∑
2l≤|a2|<2l+1
∑
a0
3∑
i=1
|σi(P )|ρ, (14)
where P (x) = a2x
2+a1x+a0. If 2
k ≤ |a1| < 2k+1 and 2l ≤ |a2| < 2l+1 then, by
(13), |σi(P )| ¿ 2−(1+ε)(k+l)−max{k,l}. Moreover, by (7), the number of different
a0 such that σ(P ) 6= ∅ is ¿ 2max{k,l}. Now it follows from (14) that∑
P∈P2
∑3
i=1 |σi(P )|ρ ¿∑∞
k=0
∑∞
l=0
∑
2k≤|a1|<2k+1
∑
2l≤|a2|<2l+1 2
max{k,l} · (2−(1+ε)(k+l)−max{k,l})ρ ¿∑∞
k=0
∑∞
l=0 2
k · 2l · 2max{k,l} · (2−(1+ε)(k+l)−max{k,l})ρ =∑∞
k=0
∑∞
l=0 2
(1−ρ(1+ε))(k+l)+(1−ρ)max{k,l} ≤∑∞
k=0
∑∞
l=0 2
(1−ρ(1+ε))(k+l)+(1−ρ)(k+l) =
∑∞
k=0
∑∞
l=0 2
(2−ρ(2+ε))(k+l) =(∑∞
k=0 2
(2−ρ(2+ε))k
)
·
(∑∞
l=0 2
(2−ρ(2+ε))l
)
.
Since ρ > 2/(2 + ε), we have 2 − ρ(2 + ε) < 0. It is now easy to see that the
sum
∑∞
l=0 2
(2−ρ(2+ε))l converges. Therefore, we have∑
P∈P2
∑3
i=1 |σi(P )|ρ <∞ (15)
for any ρ with 2/(2 + ε) < ρ < 1. By Lemma 4 in [4,pp. 94], the Hausdorff
dimension of the set consisting of x ∈ I, which belongs to infinitely many
intervals σi(P ), is at most ρ. This set is exactly M22 (ε, I). Since ρ ∈ (2/(2 +
ε), 1) is arbitrary, we have dimM22 (ε, I) ≤ 2/(2 + ε). Combining this and (11)
completes the proof of Theorem 2. 2
References
5
1. A. Baker, Transcendental number theory. Cambr. Univ. Press. 1975.
2. A. Baker and W. Schmidt, Diophantine approximation and Hausdorff dimen-
sion. Proc. London Math. Soc. 21 (1970), 1–11.
3. V.I. Bernik, Application of Hausdorff dimension in the theory of Diophantine
approximation, Acta Arithmetica. 42 (1983), 219–253, in Russian.
4. V.I. Bernik, Yu.V. Melnichuk, Diophantine approximation and Hausdorff di-
mension. Minsk 1988, in Russian.
5. V.G. Sprindzˇuk, Mahler’s problem in the metric theory of numbers, Amer.
Math. Soc., Translations of Mathematical Monographs 25 (1969).
6. V.G. Sprindzˇuk, Achievements and problems in Diophantine approximation
theory, Uspekhi Mat. Nauk 35 (1980), 3–68, english translation in Russian Math.
Surveys 35 (1980), 1–80.
7. D.Y. Kleinbock, and G.A.Margulis, Flows on homogeneous spaces and Dio-
phantine approximation on manifolds, Ann. Math. 148 (1988), 339–360.
6


Baker's conjecture and Hausdorff dimension

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Baker's conjecture and Hausdorff dimension

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