The Arithmetic of Diophantine Approximation Groups II: Mahler Arithmetic

TL;DR

This paper develops a unified algebraic framework for real numbers using polynomial diophantine approximation rings, characterizing their structure via Mahler classes and introducing an approximate ideal arithmetic based on tensor products.

Contribution

It introduces the polynomial diophantine approximation ring for real numbers and characterizes its filtration and arithmetic structure, extending the algebraic theory of Diophantine approximation.

Findings

Characterization of polynomial diophantine approximation rings by Mahler class.

Introduction of approximate ideal structures via tensor products.

Main theorem providing explicit partial product law for these groups.

Abstract

This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real number $\theta$ we associate its polynomial diophantine approximation ring: a tri-filtered subring of a nonstandard model of the ring $\mathbb{Z}[X]$. We characterize the filtration structure of the polynomial diophantine approximation ring according to the Mahler class and the Mahler type of $\theta$. The arithmetic of polynomial diophantine approximation groups is introduced in terms of the tensor product of polynomials. In particular, it is shown that polynomial diophantine approximation groups have the structure of approximate ideals: wherein a partial tensor product of two polynomial diophantine approximation groups may be performed by restriction to substructures of the tri-filtration. The explicit characterization of this partial product law is the main theorem of this paper.