Lattice point counting and height bounds over number fields and quaternion algebras

TL;DR

This paper applies lattice point counting estimates to various number theory problems involving heights over number fields and quaternion algebras, providing new bounds and existence results.

Contribution

It introduces height comparison inequalities for number fields and quaternion algebras and uses them to derive bounds and existence results for points of bounded height.

Findings

Establishes height bounds over number fields and quaternion algebras.

Provides counting estimates for integral points and units.

Derives non-commutative analogues of classical theorems like Siegel's lemma.

Abstract

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number field and over a quaternion algebra. We also show how these inequalities can be used to obtain existence results for points of bounded height over a quaternion algebra, which constitute non-commutative analogues of variations of the classical Siegel's lemma and Cassels' theorem on small zeros of quadratic forms.