Article

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

03/2008; DOI: 10.4064/aa117-1-4

Source: arXiv

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**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 pages09/2010; - [Show abstract] [Hide abstract]

**ABSTRACT:**It is shown that a non-degenerate curve in ℝ 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; - Doklady Mathematics 01/2011; 83(2):194-196. · 0.38 Impact Factor

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