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

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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 a and b greater than 1, the additive group of integers has generating sets aⁱᵢ₌₀∞ and bʲⱼ₌₀∞ with associated metrics dₐ and dB, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that aᵐ = bⁿ.