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There is a standard “word length” metric canonically associated to any set of generators for a group. In particular, for any integers and greater than , the additive group has generating sets and with associated metrics and , respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers and such that .
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