Simultaneous Diophantine approximation and asymptotic formulae on manifolds
Simultaneous Diophantine approximation and asymptotic formulae on manifolds
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.
298-316
Dodson, M.M.
a015265b-7deb-42d6-b5fe-3f797c58a7eb
Rynne, B.P.
59fe3bf6-6f7e-4987-99a1-fa8cb8b95534
Vickers, J.A.G.
719cd73f-c462-417d-a341-0b042db88634
1996
Dodson, M.M.
a015265b-7deb-42d6-b5fe-3f797c58a7eb
Rynne, B.P.
59fe3bf6-6f7e-4987-99a1-fa8cb8b95534
Vickers, J.A.G.
719cd73f-c462-417d-a341-0b042db88634
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), .
(doi:10.1006/jnth.1996.0079).
Abstract
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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Published date: 1996
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Local EPrints ID: 29146
URI: http://eprints.soton.ac.uk/id/eprint/29146
ISSN: 0022-314X
PURE UUID: d1958a5f-225e-4772-bed3-539a7dc2329a
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Date deposited: 22 Dec 2006
Last modified: 16 Mar 2024 02:34
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Author:
M.M. Dodson
Author:
B.P. Rynne
Author:
J.A.G. Vickers
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