Intersections of homogeneous Cantor sets and beta-expansions

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