The multifractal spectra of planar waiting sets in beta expansions

TL;DR

This paper determines the Hausdorff dimension of sets of pairs of numbers with prescribed waiting time indicators in beta-expansions, showing they are always of full dimension two, and explores related generalizations.

Contribution

It establishes that the waiting sets in beta-expansions have Hausdorff dimension two for all prescribed indicator pairs, extending previous understanding of multifractal spectra.

Findings

Waiting sets have Hausdorff dimension two for all indicator pairs

Results hold for any pair with 0 ≤ a ≤ b ≤ ∞

Includes generalizations of the main theorem

Abstract

Let $\beta>1$ be a real number. In this paper, the Hausdorff dimension of sets consisting of pairs of numbers with prescribed quantitative waiting time indicators in $\beta$-expansions are determined. More precisely, let $I$ be the unit interval $[0,1)$ and write $\underline{R}^\beta(x,y)$ and $\overline{R}^\beta(x,y)$ as the lower and upper quantitative waiting time indicators of $y$ by $x$ in $\beta$-expansions, respectively. Define the waiting set on the plane by \[E_\beta(a,b)=\left\{(x,y)\in I^2\colon\underline{R}^\beta(x,y)=a,\overline{R}^\beta(x,y)=b\right\}.\] where $0\leq a\leq b\leq\infty$, then the set $E_\beta(a,b)$ is always of Hausdorff dimension two for any pair of numbers $a$ and $b$. In addition, some generalizations for this result are also given in the last section.