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Simultaneous Diophantine approximation and asymptotic formulae on manifolds

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.
0022-314X
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
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), 298-316. (doi:10.1006/jnth.1996.0079).

Record type: Article

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

Identifiers

Local EPrints ID: 29146
URI: http://eprints.soton.ac.uk/id/eprint/29146
ISSN: 0022-314X
PURE UUID: d1958a5f-225e-4772-bed3-539a7dc2329a
ORCID for J.A.G. Vickers: ORCID iD orcid.org/0000-0002-1531-6273

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Date deposited: 22 Dec 2006
Last modified: 16 Mar 2024 02:34

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Contributors

Author: M.M. Dodson
Author: B.P. Rynne
Author: J.A.G. Vickers ORCID iD

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