Aproximations in Divisible Groups: Part I

This is the first in a series of papers on Dirichlet-type approximation in the setting of Cauchy sequences in normed divisible groups. In particular, we demonstrate that the concept of approximation exponents are extendable to elements belonging to the completion of a normed uniquely divisible group and other such groups that enjoy a form of divisibility. To give a measure of how “best” the approximation can be, we introduce group theoretic functions (dubbed proximity functions), which generalise the notion of the order of elements in a group. A proximity function ϱ on a group with identity e is defined by three axioms: (i) ϱ(g≠e)=ϱ(g^(-1) )>0, (ii) ϱ(gh^(-1) )≤Cϱ(g)ϱ(h) and (iii) ϱ(gh^(-1) )≤Cϱ(g) if ϱ(g)=ϱ(h), where C>0 is an absolute constant. The main result in this paper is to show that given a proximity function that is in a certain sense discontinuous at the identity, then Cauchy sequences in a uniquely divisible group G do not converge inside G; in the sequels, we consider the case of convergence inside the completion of G but not inside G.