96 research outputs found
Inhomogeneous Diophantine approximation over the field of formal Laurent series
AbstractDe Mathan [B. de Mathan, Approximations diophantiennes dans un corps local, Bull. Soc. Math. France, Suppl. Mém. 21 (1970)] proved that Khintchine's theorem on homogeneous Diophantine approximation has an analogue in the field of formal Laurent series. Kristensen [S. Kristensen, On the well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 135 (2003) 255–268] extended this metric theorem to systems of linear forms and gave the exact Hausdorff dimension of the corresponding exceptional sets. In this paper, we study the inhomogeneous Diophantine approximation over a field of formal Laurent series, the analogue Khintchine's theorem and Jarnik–Besicovitch theorem are proved
An Inhomogeneous Transference Principle and Diophantine Approximation
In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the
fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation
on manifolds. Subsequently, there has been dramatic progress in this area of
research. However, the techniques developed to date do not seem to be
applicable to inhomogeneous approximation. Consequently, the theory of
inhomogeneous Diophantine approximation on manifolds remains essentially
non-existent.
In this paper we develop an approach that enables us to transfer homogeneous
statements to inhomogeneous ones. This is rather surprising as the
inhomogeneous theory contains the homogeneous theory and so is more general. As
a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk
conjecture. Furthermore, we prove a complete inhomogeneous version of the
profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of
friendly measures. The results obtained in this paper constitute the first step
towards developing a coherent inhomogeneous theory for manifolds in line with
the homogeneous theory.Comment: 37 pages: a final section on further developments has been adde
Inhomogeneous Diophantine approximation over fields of formal power series
We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem
over fields of power series . Furthermore, we study the
approximation to a given point in by
the -orbit of a given point in
.Comment: 22 page
Diophantische Approximationen
This Number Theoretic conference was focused on the following subjects: the Littlewood conjecture, simultaneous homogeneous and inhomogeneous Diophantine approximation, geometry of numbers, irrationality, Diophantine approximation in function fields, counting questions in number fields, effective methods for resolution of Diophantine equations
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