The Multiplicative golden mean shift has infinite Hausdorff measure

TL;DR

This paper proves that the multiplicative golden mean shift, a set defined by a specific binary expansion rule, has infinite Hausdorff measure at its Hausdorff dimension, providing detailed gauge results.

Contribution

It establishes that the multiplicative golden mean shift has infinite Hausdorff measure at its dimension, extending previous dimension calculations with measure properties.

Findings

The set has infinite Hausdorff measure at its Hausdorff dimension.

Precise gauge functions for infinite measure are identified.

The result deepens understanding of measure properties of multiplicative shifts.

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.