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Journal of the London Mathematical Society Advance Access published online on March 16, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn080
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© 2009 London Mathematical Society

Ternary expansions of powers of 2

Jeffrey C. Lagarias

Department of Mathematics
The University of Michigan
530 Church Street
Ann Arbor, MI 48109-1043
USA

Erdos asked how frequently 2n 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 xn ({lambda}) = {lfloor} {lambda} 2n {rfloor}, where {lambda} >0 is a real number, along with its untruncated version, whereas the second considers 3-adic expansions of sequences yn({lambda}) = {lambda} 2n, where {lambda} 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->{infty} 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.

The author was supported by the NSF grant DMS-0500555.

Dedicated to Mel Nathanson on his 60th birthday

2000 Mathematics Subject Classification 11A63 (primary), 28A78, 37E05 (secondary).

Received May 10, 2007; revised July 23, 2008;
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