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ON A PROBLEM OF BERNIK, KLEINBOCK AND MARGULIS

Published online by Cambridge University Press:  01 August 2011

NATALIA BUDARINA
NATALIA BUDARINA*
Affiliation:
Department of Mathematics and Statistics, NUI Maynooth, Maynooth, Co. Kildare, Ireland e-mail: buda77@mail.ru
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Abstract

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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. Notices2001(9) (2001), 453). More specifically, the main result provides a probabilistic criterion for the solvability of the system of inequalities |P(x)| < Ψ1(H) and |P′(x)| < Ψ2(H) in integral polynomials P of degree ≤ n and height H, where Ψ1 and Ψ2 are fairly general error functions. The proof builds upon Sprindzuk's method of essential and inessential domains and the recent ideas of Beresnevich, Bernik and Götze (Compositio Math. 146 (2010), 1165) concerning the distribution of algebraic numbers.

Keywords

11J8311K6011J13

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011, pp. 669 - 681
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

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