Intersections of multiplicative translates of $3$-adic Cantor sets II: two infinite families

Abstract

This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}Σ_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}Σ_{3,\bar{2}}$ for integers $1\leq M_1<\cdots<M_n$, where $Σ_{3,\bar{2}}$ denotes the $3$-adic Cantor set. This study was motivated by questions concerning the discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits under this action intersect the $3$-adic Cantor set $Σ_{3,\bar{2}}$ infinitely many times. It is conjectured that it has Hausdorff dimension $0$. Part I showed that upper bounds on the Hausdorff dimension of the exceptional set can be extracted from knowing Hausdorff dimensions of sets of the kind above, in cases where all $M_i$ are powers of $2$. These intersection sets were shown to be fractals whose points have $3$-adic expansions describable by labeled paths in a finite automaton, whose Hausdorff dimension is exactly computable and is of the form $\log_3(β)$ where $β$ is a real algebraic integer. It gave algorithms for determination of the automaton, and computed examples showing that the dependence of the automaton and the value $β$ on the parameters $(M_1,\ldots,M_n)$ is complicated. The present paper studies two new infinite families of examples, illustrating interesting behavior of the automata and of the Hausdorff dimension of the associated fractals. One family has associated automata whose directed graph has a nested sequence of strongly connected components of arbitrarily large depth. The second family leads to an improved upper bound for the Hausdorff dimension of $\mathcal{E}(\mathbb{Z}_3)$ of $\log_3 ϕ\approx 0.438018$, where $ϕ$ denotes the Golden ratio.