Metric dimensions of metric spaces over integers

Abstract

We consider metric spaces over the ring of integers. For any generating set S of ℤ, it is shown that the metric dimension of the metric space X = X(ℤ, S) is not greater than 2 max S. The resolving set of metric space X = X(ℤ, S) is determined. If S = {−m, −(m − 1), . . . , −1, 1, . . . , m − 1, m}, then the metric dimension of the metric space X = X(ℤ, S) is m + 1. We determine the basis of the metric space X = X(ℤ, S).