PhD
Phis. -Math. Sci. Kovalevskaya E.I., PhD Phis. -Math. Sci.
Morozova
The zero-one
law in simultaneous inhomogeneous Diophantine approximation for the points of the
square and the cubic parabolas with respect to Arhimedean and non-Arhimedean
valuations
1.
We extend the author’ earlier result
[4] to inhomogeneous case.
Let y: N ® R+, be a monotonically decreasing function and
. (1)
Let Pn = P(y)
= anyn +…+a1y
+ a0ÎZ[y],
deg Pn = n (n = 2 or 3) and
H = h(Pn) =
. Let k ³ 2
be integer, pi³ 2
be prime numbers (1 £ i £ k), 
be the field of pi-adic numbers, 
be
the pi
-adic valuation. Suppose that
Q = R´C´
and Q1 = R´ (k ³
2). We define a measure 
in Q as a product
of the Lebesque measures m
and m¢
in R and R2 respectively, and the Haar
measure 
in (1 £ i £ k), that is, 
.We define also a measure 
in Θ1
as a product of the Lebesque measure m in R and the Haar measure in 
, that is, 
.
We consider two systems of
inequalities
(2)
where 
Θ1 and
Θ1 and
, 
and
(3)
where 
Θ and 
Θ. The parameters satisfy the following conditions: max 
min 
, 
Besides, we have the
conditions 
and 
in the case 
, and 
in the case 
. Note that the first conditions for the parameters except
the trivial inequalities in (2) and (3). We prove two theorems.
Theorem
1. For every vector
Θ1 the system of inequalities (2) has only a finite number of solutions in
polynomials P2 for almost all 
Θ1.
Theorem
2. For every vector
Θ the system of
inequalities (3) has only a finite number of solutions in polynomials P3 for
almost all
Θ.
Other words, we can say that according to the
theorem 1 for every vector Θ1 the system of inequalities (2) has
infinitely many solutions in polynomials P2
for almost none Θ1. Similarly we can get a new
formulation of the theorem 2. The
condition (1) is a crucial condition for such a metric characteristic. If the
opposite condition carries out, i. e., 
then according to the
Khintcine theorem (1924) we can expect that for every vector Θ1 the system of inequalities (2) has
infinitely many solutions in polynomials P2
for almost all Θ1 and we can have a similarly result for
(3). Thus, we see that these metric
assertions have the character of assertions of the “almost none” or “almost
all” type (so-called zero-one law). Hence, the problem under consideration
belongs to the metric theory of Diophantine approximation on manifolds which is
developed intensively this time [1-4].
Recall that E. Lutz (1955) was the first who
considered an inhomogeneous in problem
in Qp for one linear
polynomial with function 
in the right hand part
of the last inequality of (2) or (3) without the other inequalities, i. e., 
. Inhomogeneous questions are rather different in character
to the homogeneous ones because they concern the questions of how the points
are distributed rather than how the points can be closed to the integers.
Further it is well known that if 
is an irreducible
polynomial then for any two its zeros 
and 
,
, the following inequality holds 
The situation is different when we consider 
instead of where d is an irrational number. With a point
of view of the continuity it is readily proved that for any 
we can f 
where p1 and p2 are the zeros of . Besides, the zeros of are the transcendental
numbers if d is the same number.
Note that the inhomogeneous
Diophantine approximation for the Veronese curve were investigated earlier by
V. Bernik, H. Dickinson and M. Dodson [1] in only, V. Bernik, H. Dickinson and
J. Yuan [2] in 
only.
In order to prove the
theorems we need four lemmas. As in [5, p. 32, 93], the investigation of
systems (2) and (3) can be reduced to the case of the primitive irreducible
polynomials Pn with
and 
. We denote the set of those
polynomials as 
be the subset of polynomials
for which 
and H is a sufficient large integer. Let 
be the zeros of the
polynomial Pn in C and 
be the zeros of the
polynomial Pn in 
, where be the smallest field
containing 
and all algebraic numbers,
.
Lemma 1. Let 
and 
. Then
(4)
For proof see Lemma 7 [5, p.19].
Definition.
We denote the smallest m for which (4) is true by m0.
Lemma
2. Let and with 
where c is a constant depending only on n,
. Then
for every root
of Pn .
For proof see Lemma 1 [5, p.13].
Lemma
3. Let and with
where c1 is a constant depending only on n.
Then
for every root 
of Pn .
For proof see Lemma 3 [2].
Lemma 4. Let
(5)
and 
(6)
be the two systems of
inequalities with
Θ1
and 
Θ
respectively, where H, 
Θ and parameters 
are defined as in the
theorems 2, 3. Then the systems of
inequalities (5) and (6) are satisfied by at most finitely many polynomials
and 
for almost all Θ1 and
for almost all Θ respectively.
This is a
main theorem in [4].
2.
Reduction
to a polynomial
and Proof the theorems.
In
the next we use the following notation: 1) [X], the integral part of 
,2) 
is equivalent to the
simultaneous validity of 
and
.
Let 
, where m0 is fixed integer from
Definition. Consider, for example, the second inequality of (2) when i=1
. If it holds infinitely often for a set of positive measure one can be
readily verify that the set of solutions of the inequality 
also
has positive measure (see [6, Lemma 5] for details). It is easy to show that Q takes the form
(7)
where
for 
If the value of 
is very small then we
shall consider 
(or 
) instead of it. Then
the new p1-adic value
equals 1. We denote the new d1
as
. Hence, we may assume without loss of generality that 
Let 
be a set of 
for which the second
inequality in (2) holds with 
for infinitely many Pn. Let 
is the same set with 
. Let 
.It follows from here
that 
as otherwise we obtain
a contradiction (replacing 
by 
). Hence, we may assume without loss of generality that 
where c2 is a constant depending
only on n and . Therefore according to Lemma 3 the zeros of Q
are bounded. Thus, instead of second inequality of (2) we can consider
the inequality
where 
and
the zeros of 
lie in the disk. 
Note that 
is not necessarily an
integer but it is rational. Let 
and
and 
where
N depends on the height of the
polynomial Pn associated
with , i.e., 
.
Further, according to (7) we can write the
left hand parts of the other inequalities in (2) and (3) as
and
where
and. 
Note that as above we
may assume without loss of generality that 
if 
. Hence, we have 
N if the height 
is sufficient large
and 
.
If 
where N is sufficient large then N
. Therefore according to Lemma 1 we get 
. Thus, 
if H is sufficient
large. According to Lemmas 1, 2 and 3, instead of (2) and (3) we can consider
without loss of generality the system of inequalities (5) and (6) where P2 changes into 
and P3 does into 
and where 
with
and
N is sufficient
large integer and 
Now according to the lemma 4 the theorems
1,2 are proved.
ACKNOWLEDGMENT. The research was done in
the limits of the Belorussian State Programme of Fundamental researches
(Project 05-K-065).
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