**PhD
Phis. -Math. Sci. Kovalevskaya E.I., PhD Phis. -Math. Sci.**

**Morozova
****I.****M.**

*Belarus** **State** **Agrarian** **Technic** **University**, Republic **Belarus*

**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 *P _{n}* =

*H* = *h*(*P*_{n}) =_{}_{}. Let *k *³ 2
be integer, *p*_{i}³ 2
be prime numbers (1 £ *i* £ *k*), _{}be the field of *p*_{i}-adic numbers, _{} be
the *p*_{i}
-adic valuation. Suppose that

Q = R´C´_{} and Q_{1} = R´ _{} (*k *³
2). We define a measure _{}in Q as a product
of the Lebesque measures m
and m¢
in R and R^{2 }respectively, and the Haar
measure _{}_{ } in _{} (1 £ *i* £ *k*), that is, _{}.We define also a measure _{} in Θ_{1}
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 _{}Θ_{1 }and
_{}Θ_{1} 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 _{}Θ*

**Theorem
2. *** For every vector _{}Θ the system of
inequalities *(3)

Other words, we can say that according to the
theorem 1 for every vector _{}Θ_{1 }the system of inequalities (2) has
infinitely many solutions in polynomials *P*_{2}
for almost none _{}Θ_{1}. 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 _{}Θ_{1} the system of inequalities (2) has
infinitely many solutions in polynomials *P*_{2}
for almost all _{}Θ_{1} 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 *Q _{p}* for one linear
polynomial with function

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 *d* that _{} where *p*_{1} and *p*_{2} 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 *P _{n}* with

**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 *m*_{0}.

**Lemma
2.** *Let _{} and _{} with _{} where c is a constant depending only on n,_{} . Then_{} for every root_{} of P_{n} .*

For proof see Lemma 1 [5, p.13].

**Lemma
3. ***Let _{} and_{} with_{} where c*

For proof see Lemma 3 [2].

**Lemma 4. ***Let*

* _{} _{} *(5)

*and _{} *(6)

*be the two systems of
inequalities with _{} Θ*

* *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 *m _{0} *is fixed integer from
Definition. Consider, for example, the second inequality of (2) when

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 *p*_{1}-adic value
equals 1. We denote the new *d*_{1}
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 *P _{n}*. Let

_{}

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 *P _{n}* associated
with

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 *P*_{2} changes into _{} and *P*_{3} 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).

References:

1.
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_{}2. BERNIK V. – DICKINSON H. – YUAN J.: *inhomogeneous
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3.
BERESNEVICH V. – BERNIK V. – KLEINBOCK D. – MARGULIES G. A.:* Metric
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4.
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