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

Abstract

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 TN(c,m) with respect to the normalised Hausdorff measure, i.e. we compute 1/Hs(TN(c,m)2 ∫TN(c,m)2|x-y|(Hs x Hs(x,y) where s denotes the Hausdorff dimension of 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 TN(c,m) is the set of those real numbers x ε [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/Hs(T2(2,3))2 ∫T2(2,3)2 |x-y|d(Hs x Hs)(x,y) = 4444λ2 + 2071λ + 3030 / 1241λ2 + 5650λ + 8281 = 0.36610656 ..., where λ = 1.465571232 ... is the unique positive real number such that λ3 - λ2 - 1 = 0.