Magnitude of homogeneous Moran sets in the unit interval

Abstract

Magnitude, denoted by $\operatorname{Mag}(X)$, 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 $X$, $\operatorname{Mag}(tX)$ grows like $t^{\dim X}$, 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 $d_u$, writing $E_u=(E,d_u)$. We prove that the upper and lower growth exponents of $\operatorname{Mag}(tE_u)$, called the magnitude dimensions of $E_u$, coincide respectively with the upper and lower Euclidean box dimensions of $E$. In the self-similar case with constant contraction ratio $r$, we obtain $\operatorname{Mag}(tE_u)=t^s/\widetilde{p}(\log t)+o(t^s)$ as $t\to\infty$, where $s$ is the Hausdorff dimension of $E$ and $\widetilde{p}$ is a positive smooth function of period $-\log r$. The harmonic mean of the leading coefficient $1/\widetilde{p}$ is $m\log m/((m-1)Γ(s+1))$, giving a fractal analogue of Willerton's leading-order asymptotics with a log-periodic, rather than constant, coefficient.