Dodson, M.M., Rynne, B.P. and Vickers, J.A.G.
Simultaneous Diophantine approximation and asymptotic formulae on manifolds.
Journal of Number Theory, 58, (2), . (doi:10.1006/jnth.1996.0079).
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Let?(r),r=1, 2, … be a positive decreasing sequence such that ?r=1? ?(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.
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