Weak admissibility, primitivity, o-minimality, and Diophantine approximation

TL;DR

This paper extends the concept of weak admissibility for lattices to include those relevant in Diophantine approximation and algebraic number theory, providing improved counting estimates and applications to primitive points and o-minimal structures.

Contribution

It generalizes Skriganov's weak admissibility to standard lattices in number theory, deriving sharper counting estimates and applying o-minimality to describe complex sets.

Findings

Improved lattice point counting estimates for distorted boxes and non-admissible lattices.

Sharpness criterion for the error term in lattice point counting.

Application to Diophantine approximation with primitive points and o-minimal structures.

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, Erd\H{o}s, and others. Finally, we use o-minimality to describe large classes of sets