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.

A Khintchine type theorem for affine subspaces

“…Currently, from [RSS17] it is known that affine coordinate subspaces of dimension at least two, and affine coordinate subspaces of dimension one with a Diophantine restriction on the shift vector are of Khintchine type for divergence. In the convergence side of the theory, certain coordinate hyperplanes are of Khintchine type for convergence [Ram15] and affine subspaces with a Diophantine restriction on the parametrizing matrix are of Khintchine type for convergence [HL19].…”

Approximation on Affine Subspaces

“…Choosing ψ(q) = q −τ in Theorem 24, we obtain a lower bound on dim W (τ )∩L. An upper bound on dim W (τ )∩L can be obtained from Theorem 2 in [HL19], which together immediately gives the following corollary.…”

“…It is worth noting that the upper bound can also be established with the less restrictive condition that ω(ã) < n+1 τ , and is found as Corollary 4 in [HL19]. The upper bound used above is derived from Theorem 2 in [HL19], which in fact gives a condition for an affine subspace to be of strong Jarník type for convergence.…”

“…Currently, from [RSS17] it is known that affine coordinate subspaces of dimension at least two, and affine coordinate subspaces of dimension one with a Diophantine restriction on the shift vector are of Khintchine type for divergence. In the convergence side of the theory, certain coordinate hyperplanes are of Khintchine type for convergence [Ram15] and affine subspaces with a Diophantine restriction on the parametrizing matrix are of Khintchine type for convergence [HL19].…”

“…we make a full measure statement about the set of ψ-approximable points in R d intersected with an affine subspace. This theorem coupled with a theorem from [HL19] allows us to show a Hausdorff dimension result for the set of τ -approximable points intersected with an affine subspace. There is a mild Diophantine restriction on the subspaces for which this theorem applies.…”