Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds

Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds

Letψ(r),r=1, 2, … be a positive decreasing sequence such that ∑r=1∞ψ(r)kdiverges. 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[formula]which holds for almost all points (x1, …, xk)...

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Bibliographic Details
Published in:Journal of number theory Vol. 58; no. 2; pp. 298 - 316
Main Authors: Dodson, M.M., Rynne, B.P., Vickers, J.A.G.
Format: Journal Article
Language:English
Published: Elsevier Inc 01-06-1996
Online Access:Get full text
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Summary:Letψ(r),r=1, 2, … be a positive decreasing sequence such that ∑r=1∞ψ(r)kdiverges. 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[formula]which holds for almost all points (x1, …, xk) on a smoothm-dimensional submanifoldMof Rk. The manifold satisfies certain curvature conditions which entail restrictions on the codimension. This result extends the known result when the 0?points are not constrained to lie in a submanifold, (i.e., whenM=Rk) to a reasonably general class of manifolds.
ISSN:0022-314X
1096-1658
DOI:10.1006/jnth.1996.0079