The Skolem–Bang theorems in ordered fields with an IP

Abstract

This paper is concerned with the extent to which the Skolem–Bang theorems in Diophantine approximations generalize 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.