# Hausdorff dimension of frequency sets in beta-expansions

## TL;DR

This paper proves a key relationship between the Hausdorff dimensions of sets in beta-expansions and their projections, simplifies the calculation of these dimensions for frequency sets, and provides exact formulas for pseudo-golden ratios.

## Contribution

It offers a rigorous proof of a folklore result, simplifies the computation of Hausdorff dimensions for frequency sets, and derives explicit formulas for multinacci numbers.

## Key findings

- Hausdorff dimension of sets in shift space equals that of their projections
- Dimension calculation reduces to optimizing finitely many variables
- Exact formulas obtained for frequency sets of multinacci numbers

## Abstract

By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $\beta>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_\beta$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$. It has been used in some former papers without proof. Then we clarify that for calculating the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit order when the $\beta$-expansion of $1$ is finite. Concretely, it suffices to optimize a function with finitely many variables under some restrictions. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $\beta$'s, which are called pseudo-golden ratios (also called multinacci numbers).