Metric Dimensions of Metric Spaces over Vector Groups

Let ( X , ρ ) be a metric space. A subset A of X resolves X if every point x ∈ X is uniquely identified by the distances ρ ( x , a ) for all a ∈ A . The metric dimension of ( X , ρ ) is the minimum integer k for which a set A of cardinality k resolves X . We consider the metric spaces of Cayley graphs of vector groups over Z . It was shown that for any generating set S of Z , the metric dimension of the metric space X = X ( Z , S ) is, at most, 2 max S . Thus, X = X ( Z , S ) can be resolved by a finite set. Let n ∈ N with n ≥ 2 . We show that for any finite generating set S of Z n , the metric space X = X ( Z n , S ) cannot be resolved by a finite set.