The University of Michigan Combinatorics Seminar
Fall 2009
September 18, 4:10-5:00, 3866 East Hall

Ternary expansions of powers of 2

Jeffrey C. Lagarias

University of Michigan


P. Erdös raised the question whether there are infinitely many integers n such that the ternary expansion of 2n omits the digit 2. We discuss this question, and generalize it to viewing {2n : n>0} as a special case of an orbit of a dynamical system acting on the real numbers, and of a second dynamical system acting on the 3-adic integers. The set of orbits having infinitely many elements with the property above should be "small." This leads to new questions about the sizes of intersections of multiplicative translates of 3-adic Cantor sets. Their Hausdorff dimensions have a combinatorial description and are investigated experimentally. The latter reports on joint work with an REU student, Will Abram (U. Chicago).