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Ternary expansions of powers of 2
Author(s) -
Lagarias Jeffrey C.
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn080
Subject(s) - iterated function , mathematics , integer (computer science) , dimension (graph theory) , hausdorff dimension , multiplicative function , infinity , sequence (biology) , combinatorics , real number , discrete mathematics , lambda , ternary operation , upper and lower bounds , mathematical analysis , computer science , physics , biology , optics , genetics , programming language
Erdős asked how frequently 2 n has a ternary expansion that omits the digit 2. He conjectured that this holds only for finitely many values of n . We generalize this question to consider iterates of two discrete dynamical systems. The first considers truncated ternary expansions of real sequences x n (λ) = ⌊λ2 n ⌋, where λ > 0 is a real number, along with its untruncated version, whereas the second considers 3‐adic expansions of sequences y n (λ) = λ2 n , where λ is a 3‐adic integer. We show in both cases that the set of initial values having infinitely many iterates that omit the digit 2 is small in a suitable sense. For each nonzero initial value we obtain an asymptotic upper bound as k → ∞ on the number of the first k iterates that omit the digit 2. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3‐adic Cantor sets.