Intersections of homogeneous Cantor sets and beta-expansions

Abstract

Let ${\Gamma }_{\beta ,N}$ be the $N$-part homogeneous Cantor set with $\beta \in (1/(2N-1),1/N)$. Any string $(j_{\ell }{)}_{\ell =1}^{\mathbb{N}}$ with $j_{\ell }\in \{0,\pm 1,...,\pm (N-1)\}$ such that $t={\sum }_{\ell =1}^{\mathbb{N}}{j}_{\ell }{\beta }^{\ell -1}(1-\beta )/(N-1)$ is called a code of $t$. Let ${\mathcal{U}}_{\beta ,\pm N}$ be the set of $t\in [-1,1]$ having a unique code, and let ${\mathcal{S}}_{\beta ,\pm N}$ be the set of $t\in {\mathcal{U}}_{\beta ,\pm N}$ which make the intersection ${\Gamma }_{\beta ,N}\cap ({\Gamma }_{\beta ,N}+t)$ a self-similar set. We characterize the set ${\mathcal{U}}_{\beta ,\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $t\in {\mathcal{S}}_{\beta ,\pm N}$. Using techniques from beta-expansions, we show that there is a critical point ${\beta }_c\in (1/(2N-1),1/N)$, which is a transcendental number, such that ${\mathcal{U}}_{\beta ,\pm N}$ has positive Hausdorff dimension if $\beta \in (1/(2N-1),{\beta }_c)$, and contains countably infinite many elements if $\beta \in ({\beta }_c,1/N)$. Moreover, there exists a second critical point ${\alpha }_c=[N+1-\sqrt{(N-1)(N+3)}]/2\in (1/(2N-1),{\beta }_c)$ such that ${\mathcal{S}}_{\beta ,\pm N}$ has positive Hausdorff dimension if $\beta \in (1/(2N-1),{\alpha }_c)$, and contains countably infinite many elements if $\beta \in [{\alpha }_c,1/N)$.

23 pages, 4 figures

References in corpus (2)

• Combinatorics of linear iterated function systems with overlaps
• Univoque numbers and an avatar of Thue-Morse

Cited by in corpus (6)

• Hausdorff dimension of unique beta expansions
• On the continuity of the Hausdorff dimension of the univoque set
• Unique expansions and intersections of Cantor sets
• Relative bifurcation sets and the local dimension of univoque bases
• Pointwise densities of homogeneous Cantor measure and critical values
• Intersections of middle-$\alpha$ Cantor sets with a fixed translation