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WEAK ADMISSIBILITY, PRIMITIVITY, O-MINIMALITY, AND DIOPHANTINE APPROXIMATION

Part of: Geometry of numbers Noncompact transformation groups Probabilistic theory: distribution modulo $1$; metric theory of algorithms Additive number theory; partitions Diophantine approximation, transcendental number theory Model theory

Published online by Cambridge University Press:  17 April 2018

Martin Widmer
Martin Widmer*
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, U.K. email [email protected]
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Abstract

We generalize Skriganov’s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov’s celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erdős, and others. Finally, we use o-minimality to describe large classes of sets to which our counting results apply.

MSC classification

Primary: 11H06: Lattices and convex bodies 11P21: Lattice points in specified regions 11K60: Diophantine approximation 11J20: Inhomogeneous linear forms
Secondary: 03C64: Model theory of ordered structures; o-minimality 22F30: Homogeneous spaces
Type
Research Article
Information
Mathematika , Volume 64 , Issue 2 , 2018 , pp. 475 - 496
Copyright
Copyright © University College London 2018 

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