Hausdorff dimension of the multiplicative golden mean shift

Hausdorff dimension of the multiplicative golden mean shift

The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure

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 (xk) satisfies xkx2k = 0 for all k ≥ 1. 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 obta...

Hausdorff Dimension and mean porosity

Hausdorff Dimension for Fractals Invariant under the Multiplicative Integers

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences (xk) such that xkx2k = 0 for all k. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for mult...

The Transfinite Hausdorff Dimension

Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...