JournalspmVol. 68, No. 2pp. 191–203

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

  • Melvyn B. Nathanson

    Lehman College, CUNY, Bronx, USA
Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory cover

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Abstract

There is a standard “word length” metric canonically associated to any set of generators for a group. In particular, for any integers aa and bb greater than 11, the additive group Z\mathbb{Z} has generating sets {ai}i=0\{ a^i \}_{i=0}^{\infty} and {bj}j=0\{b^j\}_{j=0}^{\infty} with associated metrics dAd_A and dBd_B, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers mm and nn such that am=bna^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