Intrinsic Diophantine approximation on manifolds: General theory

Abstract

We investigate the question of how well points on a nondegenerate k k -dimensional submanifold M ⊆ R d M \subseteq \mathbb R^d can be approximated by rationals also lying on M M , establishing an upper bound on the “intrinsic Dirichlet exponent” for M 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 k and d 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 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 M which provides new insights on the local distribution of rational points on nondegenerate manifolds.

Intrinsic Diophantine Approximation on General Polynomial Surfaces

“…In the case where , this has been studied previously by Budarina et al in [5]. The special case of Veronese manifolds is covered in [9, §2]. Furthermore, in the case where the defining polynomials depend only on one variable this has been studied by Schleischitz [14].…”

“…However, some results have been obtained in special cases: the case of the circle is well understood [6], as is the case of certain polynomial curves [5]. More recently, results for spheres [13] and more general quadratic surfaces [8] as well as homogenous varieties [10] have been obtained by dynamic methods.…”

Rational approximation on quadrics: A simplex lemma and its consequences

“…On the other hand, it is shown in [14] that to prove some negative results, that is, to show that many points of X are not too close to rational points, one often does not need to know much about X. The main tool on which the argument of [14] is based is the Simplex Lemma originating in Davenport's work [10]. The version presented in [14,Lemma 4.1] is very general -it applies to any manifold embedded in R nand at the same time precise enough to yield some satisfying theorems in the case of quadric hypersurfaces.…”

“…The paper [20] studies the case X = S n−1 , the unit sphere in R n . Later in [13] the results of [20] were significantly strengthened and extended to the case of X being an arbitrary rational quadric hypersurface. An even more general framework was developed in [14].…”

“…Roughly speaking, in order to exhibit points on submanifolds X ⊂ R n which are close enough to rational points of X, one has to make use of the structure of X (indeed, in general it is not even guaranteed that X ∩ Q n is not empty). On the other hand, it is shown in [14] that to prove some negative results, that is, to show that many points of X are not too close to rational points, one often does not need to know much about X. The main tool on which the argument of [14] is based is the Simplex Lemma originating in Davenport's work [10].…”