Simultaneous approximation on affine subspaces

Abstract

We solve the convergence case of the generalized Baker-Schmidt problem for simultaneous approximation on affine subspaces, under natural diophantine type conditions. In one of our theorems, we do not require monotonicity on the approximation function. In order to prove these results, we establish asymptotic formulae for the number of rational points close to an affine subspace. One key ingredient is a sharp upper bound on a certain sum of reciprocals of fractional parts associated with the matrix defining the affine subspace.

arXiv commentTo appear in IMRN

References in corpus (9)

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Cited by in corpus (7)

• The density of rational points near hypersurfaces
• Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals
• Khintchine's theorem and Diophantine approximation on manifolds
• Integral points close to a space curve
• Counting rational points close to $p$-adic integers and applications in Diophantine approximation
• Sums of Reciprocals of Fractional Parts over Aligned Boxes
• On a generalization of a theorem of Popov