Natalia Budarina

Simultaneous diophantine approximation on polynomial curves

The Hausdorff dimension and measure of the set of simultaneously ψ-approximable points lying on integer polynomial curves is obtained for sufficiently small error functions. §

Simultaneous Diophantine approximation in the real, complex and p –adic fields.

In this paper it is shown that if the volume sum ?r = 18 ?(r) converges for a monotonic function ? then the set of points (x, z, w) ? R × C × Qp which simultaneously satisfy the inequalities |P(x)| = H-v1 ??1(H), |P(z)| = H-v2 ??2(H) and |P(w)|p = H-v3 ??3(H) with v1 + 2v2 + v3 = n - 3 and ?1 + 2?2 + ?3 = 1 for infinitely many integer polynomials P has measure zero.

On simultaneous rational approximation to a p-adic number and its integral powers

Let p be a prime number. For a positive integer n and a p -adic number ξ, let λ n (ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖ q ξ‖ p ,‖ q ξ 2 ‖ p ,…,‖ q ξ n ‖ p are all less than q −λ−1 . Here, ‖ x ‖ p denotes the infimum of | x−n | p as n runs through the integers. We study the set of values taken by the function λ n .

ON A PROBLEM OF NESTERENKO: WHEN IS THE CLOSEST ROOT OF A POLYNOMIAL A REAL NUMBER?

Let P be an integer polynomial of height H. In this article we investigate the value of w where, if |P(x)| 2n - 3, and sufficiently large H the root α1 belongs to the field of real numbers and we also bound the distance between x and α1.

ON A PROBLEM OF BERNIK, KLEINBOCK AND MARGULIS

In this paper, the Khintchine-type theorems of Beresnevich ( Acta Arith . 90 (1999), 97) and Bernik ( Acta Arith . 53 (1989), 17) for polynomials are generalised to incorporate a natural restriction on derivatives. This represents the first attempt to solve a problem posed by Bernik, Kleinbock and Margulis ( Int. Math. Res. Notices 2001 (9) (2001), 453). More specifically, the main result provides a probabilistic criterion for the solvability of the system of inequalities | P ( x )| 1 ( H ) and | P ′( x )| 2 ( H ) in integral polynomials P of degree ≤ n and height H , where Ψ 1 and Ψ 2 are fairly general error functions. The proof builds upon Sprindzuks method of essential and inessential domains and the recent ideas of Beresnevich, Bernik and Gotze ( Compositio Math . 146 (2010), 1165) concerning the distribution of algebraic numbers.