Simultaneous diophantine approximation and asymptotic formulae on manifolds

M. M. Dodson, B. P. Rynne, J. A G Vickers

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


Let ?(r), r =1, 2, ... be a positive decreasing sequence such that ?8r=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{ ||qxi||: 1 = i = k} < ?(q) which holds for almost all points (x1, ..., xk) on a smooth m-dimensional submanifold M of Rk. 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 = Rk) to a reasonably general class of manifolds. © 1996 Academic Press, Inc.

Original languageEnglish
Pages (from-to)298-316
Number of pages19
JournalJournal of Number Theory
Issue number2
Publication statusPublished - Jun 1996


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