Article

# On a theorem of V. Bernik in the metrical theory of Diophantine approximation

• ##### Victor Beresnevich
03/2008; DOI: 10.4064/aa117-1-4
Source: arXiv

ABSTRACT This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed.

0 Bookmarks
·
55 Views
• Source
##### Article: Inhomogeneous theory of dual Diophantine approximation on manifolds
[Hide abstract]
ABSTRACT: The inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The generality of the inhomogeneous aspect considered within enables us to make a new contribution even to the classical theory in R^n. Furthermore, the multivariable aspect considered within has natural applications beyond the standard inhomogeneous theory such as to Diophantine problems related to approximation by algebraic integers. Comment: 37 pages
09/2010;
• ##### Article: Diophantine approximation on non-degenerate curves with non-monotonic error function
[Hide abstract]
ABSTRACT: It is shown that a non-degenerate curve in &Ropf; n satisfies a convergent Groshev theorem with a non-monotonic error function. In other words it is shown that if a volume sum converges the set of points lying on the curve which satisfy a Diophantine condition has Lebesgue measure zero.
01/2009;
• ##### Article: Simultaneous diophantine approximations with nonmonotonic error function
Doklady Mathematics 01/2011; 83(2):194-196. · 0.38 Impact Factor

1 Download
Available from