The Multiplicative golden mean shift has infinite Hausdorff measure

Abstract

In an earlier work, joint with R. Kenyon, we computed the Hausdorff dimension of the "multiplicative golden mean shift" defined as the set of all reals in [0,1] whose binary expansion (x_k) satisfies x_k x_{2k}=0 for all k=1,2... Here we show that this set has infinite Hausdorff measure in its dimension. A more precise result in terms of gauges in which the Hausdorff measure is infinite is also obtained.

18 pages; minor revision after the referee report; to appear in Proceedings of the Conference on Fractals and Related Fields II (Porquerolles, June 2011)