2026/07/29 by Ryo Matsuda, Tomoshige Yukita
Mathematics · #math.MG #msc:51F99 #msc:28A80
14 pages
arxiv created 2026/07/29 · arxiv updated 2026/07/30
Magnitude, denoted by Mag(X), is a real-valued invariant of compact metric spaces whose large-scale growth reflects their geometry. Willerton showed that, for a compact homogeneous Riemannian manifold X, Mag(tX) grows like tdim X, with the volume of X appearing in its leading asymptotic terms. We study a homogeneous Moran Cantor set E equipped with the Euclidean metric d and with its coding ultrametric du, writing Eu=(E,du). We prove that the upper and lower growth exponents of Mag(tEu), called the magnitude dimensions of Eu, coincide respectively with the upper and lower Euclidean box dimensions of E. In the self-similar case with constant contraction ratio r, we obtain Mag(tEu)=ts/\widetildep(log t)+o(ts) as t→∞, where s is the Hausdorff dimension of E and \widetildep is a positive smooth function of period -log r. The harmonic mean of the leading coefficient 1/\widetildep is mlog m/((m-1)Γ(s+1)), giving a fractal analogue of Willerton's leading-order asymptotics with a log-periodic, rather than constant, coefficient.