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 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 vectorΘ1 the system of inequalities (2) has only a finite number of solutions in polynomials P2 for almost all Θ1.

Theorem 2. For every vectorΘ the system of inequalities (3) has only a finite number of solutions in polynomials P3 for almost allΘ.

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 ind such a number d that 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).

 

References:

1. BERNIK V.I. DICKINSON H. DODSON M.: Approximation of real numbers by integral polynomials, Dokl. Nats. Akad. Nayk Belarusi 42 (1997), 51-54.

2. BERNIK V. DICKINSON H. YUAN J.: inhomogeneous diophantine approximation on polynomials, Acta Arith. 90 (1999), 37-48.

3. BERESNEVICH V. BERNIK V. KLEINBOCK D. MARGULIES G. A.: Metric Diophantine approximation: the Khintchine-Groshev theorem for non-degenerate manifolds, Moscow Math. J. 2 (2002), 203-225.

4. KOVALEVSKAYA E. I. MOROZOVA I. M.: Zero-one law in Diophantine approximation for the points of Veroneses curve of the second and the third degrees with respect to different valuations, Materially III mizhnarod. nauk.-pract. conf. Aktualni problemi suchsnykh nauk: teoriya ta practica 2006. 16-30 chervnya 2006 roku. 22. Tekhichni nauki. Dnipropetrovsk. Nauki I osvita (2006), 14-17.

5. SPRINDZUK V. G.: Mahlers problem in metric Number Theory, Nauka i Tehnika, Minsk, 1967 (Russian).

6. BAKER A.: On the theorem of Sprindzuk, Proc. Roy. Soc. London Ser. A 242 (1966), 94-104.