96 research outputs found

    Inhomogeneous Diophantine approximation over the field of formal Laurent series

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    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

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    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

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    We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series Fq((T−1))\mathbb{F}_q((T^{-1})). Furthermore, we study the approximation to a given point y‾\underline{y} in Fq((T−1))2\mathbb{F}_q((T^{-1}))^2 by the SL2(Fq[T])SL_2(\mathbb{F}_q[T])-orbit of a given point x‾\underline{x} in Fq((T−1))2\mathbb{F}_q((T^{-1}))^2.Comment: 22 page

    Diophantische Approximationen

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    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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