Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds

✍ Scribed by M.M. Dodson; B.P. Rynne; J.A.G. Vickers

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 714 KB
• Volume: 58
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0079

✦ Synopsis

Let (r), r=1, 2, ... be a positive decreasing sequence such that r=1 (r) k diverges. Using a powerful variance argument due to Schmidt, an asymptotic formula is obtained for the number of integer solutions q of the system of Diophantine inequalities max[&qx i &:

which holds for almost all points (x 1 , ..., x k ) on a smooth m-dimensional submanifold M of R k . The manifold satisfies certain curvature conditions which entail restrictions on the codimension. This result extends the known result when the points are not constrained to lie in a submanifold, (i.e., when M=R k ) to a reasonably general class of manifolds.

1996 Academic Press, Inc. &qx i &< (q), i=1, ..., k.