Hausdorff dimension of the multiplicative golden mean shift

TL;DR

This paper calculates the Hausdorff dimension of a specific fractal set defined by a multiplicative constraint on binary expansions, revealing it is smaller than its Minkowski dimension, thus contributing to fractal geometry understanding.

Contribution

It provides the first explicit computation of the Hausdorff dimension for the multiplicative golden mean shift, a novel fractal set defined by binary multiplicative conditions.

Findings

Hausdorff dimension is strictly less than Minkowski dimension for this set

Explicit formula or value for the Hausdorff dimension is derived

The set's fractal properties are characterized in detail

Abstract

We compute 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 $k\ge 1$, and show that it is smaller than the Minkowski dimension.