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Geometric group theory and arithmetic diameter

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let λ_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute λ_A(h) and estimate its growth rate.

preprint2011arXivOpen access
Melvyn B. Nathanson
math.COmath.MGmath.GRmath.NT
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Melvyn B. Nathanson

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