Lower box dimension of infinitely generated self-conformal sets

Abstract

Let $\Lambda$ be the limit set of an infinite conformal iterated function system and let $F$ denote the set of fixed points of the maps. We prove that the box dimension of $\Lambda$ exists if and only if \[ \overline{\dim}_{\mathrm B} F\leq \max \{\dim_{\mathrm H} \Lambda, \underline{\dim}_{\mathrm B} F\}. \] In particular, this provides the first examples of sets of continued fraction expansions with restricted digits for which the box dimension does not exist. More generally, we establish an explicit asymptotic formula for the covering numbers $N_r(\Lambda)$ in terms of $\dim_{\mathrm H}\Lambda$ and the covering function $r\mapsto N_r(F)$, where $N_r(\cdot)$ denotes the least number of open balls of radius $r$ required to cover a given set. Such finer scaling information is necessary: in general, the lower box dimension $\underline{\dim}_{\mathrm B} \Lambda$ is not a function of the Hausdorff dimension of $\Lambda$ and the upper and lower box dimensions of $F$, and we prove sharp bounds for $\underline{\dim}_{\mathrm B} \Lambda$ in terms of these three quantities.