Simultaneous Diophantine approximation and asymptotic formulae on manifolds

Dodson, M.M., Rynne, B.P. and Vickers, J.A.G. (1996) Simultaneous Diophantine approximation and asymptotic formulae on manifolds. Journal of Number Theory, 58, (2), 298-316. (doi:10.1006/jnth.1996.0079).


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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 k. 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=k) to a reasonably general class of manifolds.

Item Type: Article
ISSNs: 0022-314X (print)
Related URLs:
Subjects: Q Science > QA Mathematics
Divisions: University Structure - Pre August 2011 > School of Mathematics > Applied Mathematics
ePrint ID: 29146
Date Deposited: 22 Dec 2006
Last Modified: 06 Aug 2015 02:29

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