Objective
In this project it is planned to maintain recent progress in the closely linked areas of metric Diophantine approximation and lattices by exploiting and developing advances in these theories. It is expected that advantage can be taken of the interplay between the two areas. Where appropriate, applications of these results to the theory of partial differential equations will be investigated.
The global objective is to develop a coherent body of theory for simultaneous Diophantine approximation including a Khintchine type theory and Hausdorff dimension; and to advance related topics of the distribution of rational and lattice points near manifolds in a Euclidean space. Also, applications to the problem of small denominators for the smoothness of pseudo-differential operators and non-linear perturbations will be examined using the results for Hausdorff dimension in simultaneous Diophantine approximation.
More specifically, it is planned
- to investigate the relationship between simultaneous Diophantine approximation on manifolds and the distribution of nearby rational points;
- to study the distribution of rational points and related lattice points near smooth non-degenerate manifolds;
- to investigate how the algebraic and geometric properties of a manifold can influence the distribution of rational points near that manifold;
- to investigate the lattice point problem for polyhedra from various points of view, one approach being motivated by an analogy with transcendence theory;
- to obtain upper and lower bounds for the Hausdorff dimension of the set of very well approximable points lying on a manifold;
- to investigate a small denominators problem associated with pseudo-differential operators using the results for Hausdorff dimension;
- to develop a Khintchine-type theory for simultaneous Diophantine approximation on manifolds.
The results which are expected would solve a number of important current questions in metric number theory, in the distribution of rational points and in the lattice point problem.
The results will be published as papers in mathematical journals and as pre-prints, and will be presented at seminars and conferences.
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
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Funding Scheme
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Coordinator
YO1 5DD York
United Kingdom
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.