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 Publish in OALib Journal ISSN: 2333-9721 APC: Only $99  Views Downloads  Relative Articles Intersections of multiplicative translates of$3$-adic Cantor sets II: two infinite families On Intersections of Cantor Sets: Hausdorff Measure There are no C^1-stable intersections of regular Cantor sets Affine embeddings and intersections of Cantor sets p-Adic Heisenberg Cantor sets, 2 Ampleness of intersections of translates of theta divisors in an abelian fourfold On Intersections of Cantor Sets: Self-Similarity$p\$-adic estimates for multiplicative character sums p-Adic Heisenberg Cantor sets Intersections of homogeneous Cantor sets and beta-expansions More...
Mathematics  2013

# Intersections of multiplicative translates of 3-adic Cantor sets

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Abstract:

Motivated by a question of Erd\H{o}s, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Cantor sets by powers of 2. More generally, this paper studies the structure of finite intersections of general multiplicative translates of the 3-adic Cantor set by integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3-adic integers whose 3-adic expansions have one-sided symbolic dynamics given by a finite automaton. As a consequence, the Hausdorff dimension of such a set is always of the form log(\beta) for an algebraic integer \beta. This paper gives a method to determine the automaton for given data (M_1, ..., M_n). Experimental results indicate that the Hausdorff dimension of such sets depends in a very complicated way on the integers M_1,...,M_n.

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