Submission date: 30 December 2016.

Abstract:

We generalise M. M. 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, Erdös, and others. Finally, we use o-minimality to describe large classes of sets

Mathematics Subject Classification: Primary 11H06, 11P21, 11J20 11K60, Secondary 03C64, 22F30

Keywords and phrases:

Full text arXiv 1612.09467: pdf, ps.