Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory

Abstract

There is a standard “word length” metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1 , the additive group \mathbb{Z} has generating sets \{ a^i \}_{i=0}^{\infty} and \{b^j\}_{j=0}^{\infty} with associated metrics d_A and d_B , respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that a^m = b^n .