Intersections of Multiplicative Translates of 3-Adic Cantor Sets II: Two Infinite Families

ABSTRACT

This article studies the structure of finite intersections of general multiplicative translates for integers 1 ⩽ M₁ < M₂ < ⋅⋅⋅ < M_n, in which denotes the 3-adic Cantor set (of 3-adic integers whose expansions omit the digit 2), which has Hausdorff dimension log ₃2 ≈ 0.630929. This study was motivated by questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. The exceptional set is defined to be the set of all elements of whose forward orbits under this action intersect the 3-adic Cantor set infinitely many times. It is conjectured that it has Hausdorff dimension 0. An earlier article 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 (M₁, …, M_n) is complicated. The present article 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 the exceptional set of log ₃φ ≈ 0.438018, where φ denotes the Golden ratio.

KEYWORDS:

• p-adic arithmetic
• finite automata
• graph-directed
• construction
• Hausdorff dimension

2000 AMS SUBJECT CLASSIFICATION:

• Primary 11A63
• Secondary 11K55, 37B10

Acknowledgments

We thank Yusheng Luo for an important observation on the structure of the automata for the sets P_k, incorporated in Definition 4.4 and Proposition 4.5. We also thank Eric Ragan and Eric Wert, who helped write some of the code that was used for experimental computations. We are indebted to the referee for many useful comments. W. A. thanks the University of Michigan, where much of this work was carried out. W. A. and A. B. would also like to thank Ridgeview Classical Schools, which facilitated their collaboration.

Funding

W. A. was partially supported by an NSF Graduate Research Fellowship. J. L. was supported by NSF grants DMS-1101373 and DMS-1401224. Some work of J. L. on the paper was done at ICERM, where he received support from the Clay Foundation as a Clay Senior Scholar. He thanks ICERM for support and good working conditions.