The Arithmetic of Diophantine Approximation Groups I: Linear Theory

TL;DR

This paper introduces a new algebraic framework using diophantine approximation groups within nonstandard models to unify algebraic and transcendental number theory, extending ideal-theoretic concepts.

Contribution

It develops a novel arithmetic structure for diophantine approximation groups, generalizing ideal theory to a broader algebraic setting.

Findings

Defines the structure of approximate ideals and their bi-filtration.

Extends ideal-theoretic arithmetic to diophantine approximation groups.

Provides a unified algebraic framework for algebraic and transcendental number theory.

Abstract

A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models of Dedekind domains called diophantine approximation groups. The arithmetic of diophantine approximation groups is defined in a way which extends the ideal-theoretic arithmetic of algebraic number theory, using the structure of an approximate ideal: a bi-filtration by subgroups along which partial products may be performed.