Khintchine’s theorem and Diophantine approximation on manifolds
Victor Beresnevich, Lei Yang
Abstract
In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of R n .Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of R n , which resolves a longstanding problem in the theory of Diophantine approximation.Furthermore, we refine this result using Hausdorff s-measures and consequently obtain the exact value of the Hausdorff dimension of τ -well approximable points lying on any nondegenerate submanifold for a range of Diophantine exponents τ close to 1/n.Our approach uses geometric and dynamical ideas together with a new technique of 'generic and special parts'.In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold.In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold.The latter uses a result of Bernik, Kleinbock and Margulis.