Intrinsic Diophantine approximation on manifolds: General theory

TY - JOUR

T1 - Intrinsic Diophantine approximation on manifolds

T2 - General theory

AU - Fishman, Lior

AU - Kleinbock, Dmitry

AU - Merrill, Keith

AU - Simmons, David

N1 - © 2017 American Mathematical Society. This paper is split off from arXiv:1405.7650v2

PY - 2018/1

Y1 - 2018/1

N2 - 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.

AB - 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.

KW - math.NT

U2 - 10.1090/tran/6971

DO - 10.1090/tran/6971

M3 - Article

SN - 0002-9947

VL - 370

SP - 577

EP - 599

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

ER -