JournalsjfgVol. 1, No. 4pp. 349–390

Intersections of multiplicative translates of 3-adic Cantor sets

  • William C. Abram

    Hillsdale College, USA
  • Jeffrey C. Lagarias

    University of Michigan, Ann Arbor, USA
Intersections of multiplicative translates of 3-adic Cantor sets cover
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This paper is motivated by questions concerning the discrete dynamical system on the 33-adic integers Z3\mathbb{Z}_{3} given by multiplication by 22. The exceptional set E(Z3)\mathcal{E}(\mathbb{Z}_3) is defined to be the set of all elements of Z3\mathbb{Z}_3 whose forward orbits under this action intersect the 33-adic Cantor set Σ3,2ˉ\Sigma_{3, \bar{2}} (of 33-adic integers whose expansions omit the digit 22) infinitely many times. It has been shown that this set has Hausdorff dimension at most 12\frac{1}{2}, and it is conjectured that it has Hausdorff dimension 00. Upper bounds on its Hausdorff dimension can be obtained with sufficient knowledge of Hausdorff dimensions of intersections of multiplicative translates of Cantor sets by powers of 22. This paper studies more generally the structure of finite intersections of general multiplicative translates S=Σ3,2ˉ1M1Σ3,2ˉ1MnΣ3,2ˉS= \Sigma_{3, \bar{2}} \cap \frac{1}{M_1} \Sigma_{3, \bar{2}} \cap \dots \cap \frac{1}{M_n} \Sigma_{3, \bar{2}} by integers 1<M1<M2<<Mn1 < M_1 < M_2 < \dots < M_n. These sets are describable as sets of 33-adic integers whose 33-adic expansions have one-sided symbolic dynamics given by a finite automaton. This paper gives a method to determine the automaton for given data (M1,,Mn)(M_1, \dots, M_n) and to compute the Hausdorff dimension, which is always of the form log3(β)\log_{3}(\beta) where β\beta is an algebraic integer. Computational examples indicate that in general the Hausdorff dimension of such sets depends in a very complicated way on the integers M1,,MnM_1, \dots, M_n. Exact answers are obtained for certain infinite families, which show as a corollary that a relaxed notion of generalized exceptional set has a positive Hausdorff dimension.

Cite this article

William C. Abram, Jeffrey C. Lagarias, Intersections of multiplicative translates of 3-adic Cantor sets. J. Fractal Geom. 1 (2014), no. 4, pp. 349–390

DOI 10.4171/JFG/11