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

### Melvyn B. Nathanson

Lehman College, CUNY, Bronx, USA

## 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 $a^m = b^n$.

## Cite this article

Melvyn B. Nathanson, Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory. Port. Math. 68 (2011), no. 2, pp. 191–203

DOI 10.4171/PM/1888