Projects per year
Abstract
In a landmark paper ('Flows on homogeneous spaces and Diophantine approximation on manifolds', Ann. of Math. (2) 148 (1998), 339360.) Kleinbock and Margulis established the fundamental BakerSprindzuk 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 nonexistent.
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 BakerSprindzuk 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), 479523.) 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.
Original language  English 

Pages (fromto)  821851 
Number of pages  31 
Journal  Proceedings of the London Mathematical Society 
Volume  101 
Issue number  3 
DOIs  
Publication status  Published  Nov 2010 
Keywords
 PLANAR CURVES
 HAUSDORFF DIMENSION
 HOMOGENEOUS SPACES
 CONVERGENCE
 POLYNOMIALS
 MANIFOLDS
 THEOREM
 SUBSPACES
 NUMBERS
 FLOWS
Projects
 3 Finished

Classical metric Diophantine approximation revisited
24/03/08 → 23/07/11
Project: Research project (funded) › Research

Inhomogenous approximation on manifolds
15/02/08 → 14/04/11
Project: Research project (funded) › Research

Geometrical, dynamical and transference principles in nonlinear Diophantine approximation and applications
1/10/05 → 30/09/10
Project: Research project (funded) › Research