An inhomogeneous transference principle and Diophantine approximation

作者：Beresnevich Victor^*; Velani Sanju

来源：Proceedings of the London Mathematical Society, 2010, 101: 821-851.

DOI：10.1112/plms/pdq002

In a landmark paper ('Flows on homogeneous spaces and Diophantine approximation on manifolds', Ann. of Math. (2) 148 (1998), 339-360.) Kleinbock and 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 and Weiss ('On fractal measures and Diophantine approximation', Selecta Math. (N.S.) 10 (2004), 479-523.) 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.

• 出版日期2010-11