The Skolem-Bang Theorems in Ordered Fields with an $IP$

TL;DR

This paper explores the generalization of Skolem-Bang theorems in Diophantine approximation from real numbers to ordered fields with an integer part, establishing some results unconditionally and extending classical theorems.

Contribution

It demonstrates that certain Skolem-Bang theorems hold in ordered fields with an integer part and extends Dirichlet's theorem to fields with an $IE_1$ integer part.

Findings

Some Skolem-Bang theorems hold unconditionally in ordered fields with an integer part.

Results based on Dirichlet's and Kronecker's theorems are established.

Dirichlet's theorem is extended to ordered fields with $IE_1$ integer part.

Abstract

This paper is concerned with the extent to which the Skolem-Bang theorems in Diophantine approximations generalise from the standard setting of $<R,Z>$ to structures of the form $<F,I>$, where $F$ is an ordered field and $I$ is an integer part of $F$. We show that some of these theorems are hold unconditionally in general case (ordered fields with an integer part). The remainder results are based on Dirichlet's and Kronecker's theorems. Finally we extend Dirichlet's theorem to ordered fields with $IE_1$ integer part.