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 $a^m = b^n$.