Projects per year
Abstract
Let A(n,m)(psi) denote the set of psiapproximable points in Rmn. Under the assumption that the approximating function psi is monotonic, the classical KhintchineGroshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous DuffinSchaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical KhintchineGroshev theorem. This settles a multidimensional analog of Catlin's conjecture.
Original language  English 

Pages (fromto)  6986 
Number of pages  18 
Journal  International Mathematics Research Notices 
Volume  2010 
Issue number  1 
DOIs  
Publication status  Published  2010 
Keywords
 SCHAEFFER CONJECTURE
 DUFFIN
 LAWS
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