An Inhomogeneous Transference Principle and Diophantine Approximation

Abstract

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.

37 pages: a final section on further developments has been added

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• S-Arithmetic Khintchine-Type Theorem
• On transfer inequalities in Diophantine approximation
• On transfer inequalities in Diophantine approximation, II
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Cited by in corpus (16)

• The distribution of close conjugate algebraic numbers
• Degrees of freedom in vector interference channels
• Report on some recent advances in Diophantine approximation
• Higher-rank Bohr sets and multiplicative diophantine approximation
• On Diophantine transference principles
• Hausdorff theory of dual approximation on planar curves
• Zero-one laws in simultaneous and multiplicative Diophantine approximation
• Littlewood and Duffin--Schaeffer-type problems in diophantine approximation
• An Inhomogeneous Jarník type theorem for planar curves
• Systems of small linear forms and Diophantine approximation on manifolds
• Metric Diophantine approximation for systems of linear forms via dynamics
• Diophantine approximation on subspaces of ${\mathbb{R}}^n$ and dynamics on homogeneous spaces
• Inhomogeneous Diophantine approximation on planar curves
• Diophantine transference principle over function fields
• Inhomogeneous theory of dual Diophantine approximation on manifolds
• Inhomogeneous Diophantine approximation on curves and Hausdorff dimension