Intrinsic Diophantine approximation on manifolds: General theory

TL;DR

This paper develops a new theoretical framework for intrinsic Diophantine approximation on manifolds, establishing bounds on approximation quality and analyzing the measure and dimension of sets of well and badly approximable points.

Contribution

It introduces an explicit, optimal bound on the intrinsic Dirichlet exponent for nondegenerate manifolds, a novel approach for rationals on manifolds, and a new Simplex Lemma analogue.

Findings

Set of badly intrinsically approximable points has full dimension.

Set of very well intrinsically approximable points has zero measure.

Bound on the intrinsic Dirichlet exponent is explicit and optimal in several cases.

Abstract

We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent" for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on $M$ distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the Simplex Lemma for rationals lying on $M$ which provides new insights on the local distribution of rational points on nondegenerate manifolds.