Diophantine Approximation Groups, Kronecker Foliations and Independence

TL;DR

This paper introduces diophantine approximation groups and Kronecker foliations to provide new algebraic and geometric characterizations of dependence, offering reformulations of key conjectures and theorems in transcendence theory.

Contribution

It presents novel algebraic and geometric frameworks for understanding dependence and reformulates major conjectures using these structures.

Findings

Reformulation of Baker and Lindemann-Weierstrass theorems

New algebraic and geometric characterizations of dependence

Connection of diophantine approximation groups to model theory

Abstract

We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of $K$-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geometric (graph) rigidities -- of the Theorems of Baker and Lindemann-Weierstrass, the Logarithm Conjecture and the Schanuel Conjecture. There is an Appendix describing diophantine approximation groups as model theoretic types.