Average distances between points in graph-directed self-similar fractals

We study several distinct notions of average distances between points belonging to graph-directed self-similar subsets of R. In particular, we compute the average distance with respect to graph-directed self-similar measures, and with respect to the normalised Hausdorff measure. As an application of our main results, we compute the average distance between two points belonging to the Drobot-Turner set T-N(c, m) with respect to the normalised Hausdorff measure, i.e. we compute 1/H-s(T-N(c, m))(2) integral(TN(c, m)2) vertical bar x-y vertical bar d(H-s x H-s)(x, y), where s denotes the Hausdorff dimension of T-N(c, m) and H-s is the s-dimensional Hausdorff measure; here the Drobot-Turner set (introduced by Drobot & Turner in 1989) is defined as follows, namely, for positive integers N and m and a positive real number c, the Drobot-Turner set T-N(c, m) is the set of those real numbers x is an element of[0, 1] for which any m consecutive base N digits in the N-ary expansion of x sum up to at least c. For example, if N = 2, m = 3 and c = 2, then our results show that 1/H-s(T-2(2, 3))(2) integral(T2(2,3)2) vertical bar x-y vertical bar d(H-s x H-s)(x, y) = 4444 lambda(2) + 2071 lambda + 3030/12141 lambda(2) + 5650 lambda + 8281 = 0.36610656..., where lambda = 1.465571232 ... is the unique positive real number such that lambda(3) - lambda(2) - 1 = 0.