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arXiv:0802.1910v1 [math.NT] 13 Feb 2008
On a theorem of V. Bernik in the metrical theory of
Diophantine approximation
by
V. Beresnevich (Minsk)1
1. Introduction.
denote the number of elements in a finite set S; the Lebesgue measure
of a measurable set S ⊂ R will be denoted by |S|; Pn will be the set of
integral polynomials of degree ? n.
denote the height of P, i.e. the maximum of the absolute values of its
coefficients; Pn(H) = {P ∈ Pn: H(P) = H}. The symbol of Vinogradov
≪ in the expression A ≪ B means A ? CB, where C is a constant. The
symbol ≍ means both ≪ and ≫. Given a point x ∈ R and a set S ⊂ R,
dist(x,S) = inf{|x−s| : x ∈ S}. Throughout, Ψ will be a positive function.
Mahler’s problem. In 1932 K. Mahler [10] introduced a classification
of real numbers x into the so-called classes of A,S,T and U numbers ac-
cording to the behavior of wn(x) defined as the supremum of w > 0 for
which
|P(x)| < H(P)−w
We begin by introducing some notation: #S will
Given a polynomial P, H(P) will
holds for infinitely many P ∈ Pn. By Minkovski’s theorem on linear forms,
one readily shows that wn(x) ? n for all x ∈ R. Mahler [9] proved that
for almost all x ∈ R (in the sense of Lebesgue measure) wn(x) ? 4n, thus
almost all x ∈ R are in the S-class. Mahler has also conjectured that for
almost all x ∈ R one has the equality wn(x) = n. For about 30 years the
progress in Mahler’s problem was limited to n = 2 and 3 and to partial
results for n > 3. V. Sprindzuk proved Mahler’s conjecture in full (see [12]).
A. Baker’s conjecture. Let Wn(Ψ) be the set of x ∈ R such that there
are infinitely many P ∈ Pnsatisfying
(1)
|P(x)| < Ψ(H(P)).
1The work has been supported by EPSRC grant GR/R90727/01
Key words and phrases: Diophantine approximation, Metric theory of Diophantine
approximation, the problem of Mahler
2000 Mathematics Subject Classification: 11J13, 11J83, 11K60
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A. Baker [1] has improved Sprindˇ zuk’s theorem by showing that
|Wn(Ψ)| = 0 if
∞
?
h=1
Ψ1/n(h) < ∞ and Ψ is monotonic.
He has also conjectured a stronger statement proved by V. Bernik [4] that
|Wn(Ψ)| = 0 if the sum
(2)
∞
?
h=1
hn−1Ψ(h)
converges and Ψ is monotonic. Later V. Beresnevich [5] has shown that
|R \ Wn(Ψ)| = 0 if (2) diverges and Ψ is monotonic. We prove
Theorem 1. Let Ψ : R → R+be arbitrary function (not necessarily
monotonic) such that the sum (2) converges. Then |Wn(Ψ)| = 0.
Theorem 1 is no longer improvable as, by [5], the convergence of (2) is
crucial. Notice that for n = 1 the theorem is simple and known (see, for
example, [8, p.121]). Therefore, from now on we assume that n ? 2.
2. Subcases of Theorem 1.
sets denoted by Wbig(Ψ), Wmed(Ψ) and Wsmall(Ψ) consisting of x ∈ R such
that there are infinitely many P ∈ Pnsimultaneously satisfying (1) and one
of the following inequalities
Let δ > 0. We define the following 3
(3)1 ? |P′(x)|,
(4)H(P)−δ? |P′(x)| < 1,
(5)
|P′(x)| < H(P)−δ
respectively. Obviously Wn(Ψ) = Wbig(Ψ)∪Wmed(Ψ)∪Wsmall(Ψ). Hence to
prove Theorem 1 it suffices to show that each of the sets has zero measure.
Since sum (2) converges, Hn−1Ψ(H) tends to 0 as H → ∞. Therefore,
(6) Ψ(H) = o(H−n+1) as H → ∞.
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3. The case of a big derivative. The aim of this section is to prove
that |Wbig(Ψ)| = 0. Let Bn(H) be the set of x ∈ R such that there exists a
polynomial P ∈ Pn(H) satisfying (3). Then
(7)Wbig(Ψ) =
∞ ?
N=1
∞ ?
H=N
Bn(H).
Now |Wbig(Ψ)| = 0 if |Wbig(Ψ)∩I| = 0 for any open interval I ⊂ R satisfying
(8)0 < c0(I) = inf{|x| : x ∈ I} < sup{|x| : x ∈ I} = c1(I) < ∞.
Therefore we can fix an interval I satisfying (8).
By (7) and the Borel-Cantelli Lemma, |Wbig(Ψ) ∩ I| = 0 whenever
(9)
∞
?
H=1
|Bn(H) ∩ I| < ∞.
By the convergence of (2), condition (9) will follow on showing that
(10)
|Bn(H) ∩ I| ≪ Hn−1Ψ(H)
with the implicit constant in (10) independent of H.
Given a P ∈ Pn(H), let σ(P) be the set of x ∈ I satisfying (3). Then
(11)Bn(H) ∩ I =
?
P∈Pn(H)σ(P).
Lemma 1. Let I be an interval with endpoints a and b. Define the following
sets I′′= [a,a + 4Ψ(H)] ∪ [b − 4Ψ(H),b]
sufficiently large H for any P ∈ Pn(H) such that σ(P) ∩ I′?= ∅, for any
x0∈ σ(P) ∩ I′there exists α ∈ I such that P(α) = 0, |P′(α)| > |P′(x0)|/2
and |x0− α| < 2Ψ(H)|P′(α)|−1.
and
I′= I \ I′′. Then for all
The proof of this Lemma nearly coincides with the one of Lemma 1 in
[5] and is left for the reader. There will be some changes to constants and
notation and one also will have to use (6).
Given a polynomial P ∈ Pn(H) and a real number α such that P′(α) ?=
0, define σ(P;α) = {x ∈ I : |x − α| < 2Ψ(H)|P′(α)|−1}. Let I′and I′′be
defined as in Lemma 1. For every polynomial P ∈ Pn(H), we define the set
ZI(P) = {α ∈ I : P(α) = 0 and |P′(α)| ? 1/2}.
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By Lemma 1, for any P ∈ Pn(H) we have the inclusion
(12)σ(P) ∩ I′⊂
?
α∈ZI(P)σ(P;α).
Given k ∈ Z with 0 ? k ? n, define
Pn(H,k) = {P = anxn+ ··· + a0∈ Pn(H) : ak= 0}
and for R ∈ Pn(H,k) let Pn(H,k,R) = {P ∈ Pn(H) : P − R = akxk}. It is
easily observed that
(13)Pn(H) =
n?
k=0
?
R∈Pn(H,k)Pn(H,k,R)
and
(14)#Pn(H,k) ≪ Hn−1
for every k.
Taking into account (11), (13), (14) and that |I′′| ≪ Ψ(H), it now becomes
clear that to prove (10) it is sufficient to show that for every fixed k and
fixed R ∈ Pn(H,k)
(15)
???
?
P∈Pn(H,k,R)
σ(P) ∩ I′??? ≪ Ψ(H).
Let k and R be fixed. Define the rational function˜R(x) = x−kR(x).
By (8), there exists a collection of intervals [wi−1,wi) ⊂ I (i = 1,...,s),
which do not intersect pairwise and cover I, such that˜R(x)′is monotonic
and does not change the sign on every interval [wi−1,wi). It is clear that s
depends on n only. Let ZI,R=?
and ZI,R∩ [wi−1,wi) = {α(1)
i
Pn(H,k,R), we obviously have the identity
P∈Pn(H,k,R)ZI(P), ki= #(ZI,R∩[wi−1,wi))
i,...,α(ki)
}, where α(j)
i
< α(j+1)
i
. Given a P ∈
xkP′(x) − kxk−1P(x)
x2k
=
?P(x)
xk
?′
=˜R(x)′.
Taking x to be α ∈ ZI(P) leads to
Now, by Lemma 1, |σ(P;α)| ≪ Ψ(H)|P′(α)|−1≪ Ψ(H)|˜R′(α)|−1.
P′(α)
αk
=˜R(α)′. By (8), |P′(α)| ≍ |˜R(α)′|.
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Using (12), we get
???
?
P∈Pn(H,k,R)
σ(P) ∩ I′??? ≪ Ψ(H)
s
?
i=1
ki
?
j=1
1
|˜R(α(j)
i)|
Now to show (15) it suffices to prove that for every i (1 ? i ? s)
(16)
ki
?
j=1
|˜R′(α(j)
i)|−1≪ 1.
Fix an index i (1 ? i ? s). If ki? 2 then we can consider two sequen-
tial roots α(j)
i
and α(j+1)
i
of two rational functions˜R + ai,j
respectively. For convenience let us assume that˜R′is increasing and posi-
tive on [wi−1,wi). Then˜R is strictly monotonic on [wi−1,wi), and we have
ai,j
k
?= ai,j+1
k
. It follows that |ai,j
Theorem and the monotonicity of˜R′, we get
k and˜R + ai,j+1
k
k− ai,j+1
k
| ? 1. Using The Mean Value
1 ? |ai,j
0− ai,j+1
0
| = |˜R′(α(j)
i) −˜R′(α(j+1)
i
)| =
= |˜R′(˜ α(j)
i)| · |α(j)
i
− α(j+1)
i
| ? |˜R′(α(j+1)
i
)| · |α(j)
i
− α(j+1)
i
|,
where ˜ α(j)
α(j+1)
i
i
is a point between α(j)
− α(j)
i
and α(j+1)
i
. This implies |˜R′(α(j+1)
i
)|−1?
i, whence we readily get
ki−1
?
j=1
|˜R′(α(j+1)
i
)|−1?
ki−1
?
j=1
?
α(j+1)
i
− α(j)
i
?
= α(ki)
i
− α(1)
i
? wi− wi−1.
The last inequality and |˜R′(α(1)
verified that (16) holds for every i with ki? 2 and is certainly true when
ki= 1 or ki= 0. This completes the proof of the case of a big derivative.
i)| ≍ |P′(α(1)
i)| ≫ 1 yield (16). It is easily
4. The case of a medium derivative.
val I satisfying (8). The statement |Wmed(Ψ)| = 0 will now follow from
|Wmed(Ψ) ∩ I| = 0. We will use the following
As above we fix an inter-
Lemma 2 (see Lemma 2 in [6]). Let α0,...,αk−1,β1,...,βk∈ R?{+∞}
be such that α0> 0, αj> βj? 0 for j = 1,...,k − 1 and 0 < βk< +∞.
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Let f : (a,b) → R be a C(k)function such that infx∈(a,b)|f(k)(x)| ? βk. Then,
the set of x ∈ (a,b) satisfying
?
|f(x)| ? α0,
βj ? |f(j)(x)| ? αj (j = 1,...,k − 1)
is a union of at most k(k + 1)/2 + 1 intervals with lengths at most
min0?i<j?k3(j−i+1)/2(αi/βj)1/(j−i). Here we adopt
c
0= +∞ for c > 0.
Given a polynomial P ∈ Pn(H), we redefine σ(P) to be the set of solu-
tions of (4). Since P(n)(x) = n!an, we can apply Lemma 2 to P with k = n
and
α0= Ψ(H), α1= 1, β1= inf
x∈σ(P)|P′(x)| ? H−δ, βn= 1,
α2= ··· = αn−1= +∞, β2= ··· = βn−1= 0.
Then we conclude that σ(P) is a union of at most n(n + 1)/2 + 1 intervals
of length ≪ α0/β1. There is no loss of generality in assuming that the
sets σ(P) are intervals as, otherwise, we would treat the intervals of σ(P)
separately. We also can ignore those P for which σ(P) is empty. For every
P we also define a point γP∈ σ(P) such that infx∈σ(P)|P′(x)| ?1
The existence is easily seen. Now we have
2|P′(γP)|.
(17)
|σ(P)| ≪ Ψ(H)|P′(γP)|−1.
It also follows from the choice of γP that
(18)H(P)−δ? |P′(γP)| < 1.
Now define expansions of σ(P) as follows:
σ1(P) := {x ∈ I : dist(x,σ(P)) < (H|P′(γP)|)−1},
σ2(P) := {x ∈ I : dist(x,σ(P)) < H−1+2δ}.
By (4), σ1(P) ⊂ σ2(P). Moreover, it is easy to see that
(19)σ1(P) ⊂ σ2(Q) for any Q ∈ Pn(H) with σ1(Q) ∩ σ1(P) ?= ∅.
It is readily verified that |σ1(P)| ≍ (H|P′(γP)|)−1, and therefore, by
(17),
|σ(P)| ≪ |σ1(P)|HΨ(H).
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Take any x ∈ σ2(P). Using the Mean Value Theorem, (18) and |x −
γP| ≪ H−1+2δ, we get |P′(x)| ? |P′(γP)| + |P′′(˜ x)((x − γP)| ≪ 1 + H ·
H−1+2δ≪ H2δ, where ˜ x is between x and γP. Similarly we estimate |P(x)|
resulting in
(20)
|P(x)| ≪ H−1+4δ, |P′(x)| ≪ H2δfor any x ∈ σ2(P).
Now for every pair (k,m) of integers with 0 ? k < m ? n we define
Pn(H,k,m) = {R = anxn+ ··· + a0∈ Pn(H) : ak= am= 0}
and for a given polynomial R ∈ Pn(H,k,m) we define
Pn(H,k,m,R) = {P = R + amxm+ akxk∈ Pn(H)}.
The intervals σ(P) will be divided into 2 classes of essential and non-
essential intervals. The interval σ(P) will be essential if for any choice of
(k,m,R) such that P ∈ Pn(H,k,m,R) for any Q ∈ Pn(H,k,m,R) other
than P we have σ1(P) ∩ σ1(Q) = ∅. For fixed k, m and R summing the
measures of essential intervals gives
?
|σ(P)| ? HΨ(H)
?
|σ1(P)| ? HΨ(H)|I| ≪ HΨ(H).
As #Pn(H,k,m) ≪ Hn−2and there are only n(n + 1)/2 different pairs
(k,m) we obtain the following estimate
?
essential intervals σ(P) with P∈Pn(H)
|σ(P)| ≪ Hn−1Ψ(H).
Thus, by the Borel-Cantelli Lemma and the convergence of (2), the set of
points x of Wmed(Ψ) ∩ I which belong to infinitely many essential intervals
is of measure zero.
Now let σ(P) be non-essential.
is a choice of k,m,R such that P ∈ Pn(H,k,m,R) and there is a Q ∈
Pn(H,k,m,R) different from P such that
Then, by definition and (19) there
σ(P) ⊂ σ1(P) ⊂ σ2(P) ∩ σ2(Q).
On the set σ2(P)∩σ2(Q) both P and Q satisfy (20) and so does the difference
P(x)−Q(x) = bmxm+bkxk. It is not difficult to see that bm?= 0 if H is big
enough. Therefore using (20)we get
(21)
???xm−k+bk
bm
??? ≪H−1+4δ
|bm|
? H−1+4δ
andmax{|bm|,|bk|} ≪ H2δ.
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Now let x belongs to infinitely many non-essential intervals. Without loss
of generality we assume that x is transcendental as otherwise it belongs to
a countable set, which is of measure zero. Therefore (21) is satisfied for
infinitely many bm,bk∈ Z. Hence, the inequality
???xm−k−p
q
??? < q−1−5δ
2δ
holds for infinitely many p,q ∈ Z. Taking δ =
and applying standard Borel-Cantelli arguments (see [8, p.121]) we complete
the proof of the case of a medium derivative for non-essential intervals.
1
10so that1−5δ
2δ
becomes 2+δ
5. The case of a small derivative.
|Wsmall(Ψ)| = 0. We will make use of Theorem 1.4 in [7]. By taking d = 1,
f = (x,x2,...,xn), U = R, T1= ... = Tn= H, θ = H−n+1, K = H−δin
that theorem, we arrive at
In this section we prove that
Theorem 2. Let x0 ∈ R and δ′=
interval I0⊂ R containing x0and a constant E > 0 such that
min(δ,n−1)
(n+1)(2n−1). Then there exists a finite
???
In particular Theorem 2 implies that for any δ > 0 the set of x ∈ R, for
which there are infinitely many polynomials P ∈ Pnsatisfying the system
?
P∈Pn,0<H(P)?H
?
x ∈ I0: |P(x)| < H−n+1,|P′(x)| < H−δ???? ? EH−δ′.
(22)
|P(x)| < H(P)−n+1, |P′(x)| < H(P)−δ,
has zero measure. Indeed, this set consists of points x ∈ I0which belong to
infinitely many sets
τm= {x ∈ I0: (22) holds for some P ∈ Pnwith 2m−1< H(P) ? 2m}
By Theorem 2, |τm| ≪ 2−mδ′with δ′> 0. Therefore,?∞
the Borel-Cantelli Lemma completes the proof of the claim.
Taking into account (6), this completes the proof of the case of a small
derivative and the proof of Theorem 1.
m=1|τm| < ∞ and
6. Concluding remarks.
sumed to be irreducible over Q and primitive (i.e. with coprime coefficients)
can also be sought. To make it more precise, let P∗
An analogue of Theorem 1 when P is as-
n(H) be the subset of
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Pn(H) consisting of primitive irreducible polynomials P of degree degP = n
and height H(P) = H. Now the set of primitive irreducible polynomials of
degree n is P∗
there are infinitely many P ∈ P∗satisfying (1).
n=?∞
H=1P∗
n(H). Let W∗
n(Ψ) be the set of x ∈ R such that
Theorem 3. Let Ψ : R → R+be arbitrary function such that the sum
(23)
∞
?
H=1
#P∗
n(H)
H
Ψ(H)
converges. Then |W∗
n(Ψ)| = 0.
For n = 1 the proof of Theorem 3 is a straightforward application of
the Borel-Cantelli Lemma and we again refer to [8, p.121]. For n > 1
the proof follows from the following 2 observations: 1) W∗
and 2) #P∗
which now implies 0 ? |W∗
#P∗
easily estimate the number of primitive reducible polynomials in Pn and
take them off the set of all primitive polynomials in Pnwhich is well known
to contain at least constant ×#Pn(H).
The Duffin-Schaeffer conjecture. The conjecture states that for n = 1 if
(23) diverges then |R \ W∗
becomes ≍ ϕ(H), where ϕ is the Euler function.
The following problem can be regarded as the generalization of the
Duffin-Schaeffer conjecture for integral polynomials of higher degree:
n(Ψ) ⊂ Wn(Ψ)
n(H) ≍ Hn. The second one guarantees the converges of (2),
n(Ψ)| ? |Wn(Ψ)| = 0. The proof of the relation
n(H) ≍ Hnis elementary and is left for the reader. In fact, one can
n(Ψ)| = 0. The multiple #P∗
1(H) in sum (23)
Prove that |R \ W∗
n(Ψ)| = 0 whenever (23) diverges.
Alternatively, for n > 1 one might investigate the measure of R\Wn(Ψ).
So far it is unclear if for n > 1 |R\W∗
0, which is another intricate question.
A remark on manifolds. In the metric theory of Diophantine approxima-
tion on manifolds one usually studies sets of Ψ-approximable points lying on
a manifold with respect to the measure induced on that manifold. Mahler’s
problem and its generalisations can be regarded as Diophantine approxima-
tion on the Veronese curve (x,x2,...,xn).
A point f ∈ Rnis called Ψ-approximable if
n(Ψ)| = 0 is equivalent to |R\Wn(Ψ)| =
?a · f? < Ψ(|a|∞),
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for infinitely many a ∈ Zn, where |a|∞= max1?i?n|ai| for a = (a1,...,an),
?x? = min{|x − z| : z ∈ Z} and Ψ : R → R+.
Let f : U → Rnbe a map defined on an open set U ⊂ Rd. We say
that f is non-degenerate at x0∈ U if for some l ∈ N the map f is l times
continuously differentiable on a sufficiently small ball centered at x0 and
there are n linearly independent over R partial derivatives of f at x0 of
orders up to l. We say that f is non-degenerate if it is non-degenerate
almost everywhere on U. The non-degeneracy of a manifold is naturally
defined via the non-degeneracy of its local parameterisation.
In 1998 D. Kleinbock and D. Margulis proved the Baker-Sprindˇ zuk
conjecture by showing that any non-degenerate manifold is strongly ex-
tremal. In particular, this implies an analogue of Mahler’s problem for
non-degenerate manifolds.A few years later an analogue of A. Baker’s
conjecture with monotonic Ψ (normally called a Groshev type theorem for
convergence) has independently been proven by V. Beresnevich [6] and by
V. Bernik, D. Kleinbock and G. Margulis [7] for non-degenerate manifolds.
It is also remarkable that the proofs were given with different methods. The
divergence counterpart (also for monotonic Ψ) has been established in [3].
In [7] a multiplicative version of the Groshev type theorem for convergence
has also been given.
Theorem 1 of this paper can be readily generalised for non-degenerate
curves: Given a non-degenerate map f : I → Rndefied on an interval I, for
any function Ψ : R → R+such that the sum (2) converges for almost all
x ∈ I the point f(x) is not Ψ-approximable. Even further, using the slicing
technique of Pyartly [11] one can extend this for a class of n-differentiable
non-degenerate manifolds which can be foliated by non-degenerate curves.
In particular, this class includes arbitrary non-degenerate analytic manifold.
However with the technique in our disposal we are currently unable to prove
the following
Conjecture. Let f : U → Rnbe a non-degenerate map, where U is an
open subset of Rd. Then for any function Ψ : R → R+such that the sum
(2) converges for almost all x ∈ U the point f(x) is not Ψ-approximable.
References
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(1966), 92–104.
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[2], A concise introduction to number theory, Cambridge Univer-
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[3] V.V. Beresnevich, V.I. Bernik, D.Y. Kleinbock, and G.A. Margulis,
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Institute of Mathematics
Academy of Sciences of Belarus
220072, Surganova 11, Minsk, Belarus
beresnevich@im.bas-net.by
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