Intersections of homogeneous Cantor sets and beta-expansions

Abstract

Let Γβ,N be the N-part homogeneous Cantor set with β ∊ (1/(2N − 1), 1/N). Any string with jℓ ∊ {0, ±1, …, ±(N − 1)} such that is called a code of t. Let be the set of t ∊ [−1, 1] having a unique code, and let be the set of which makes the intersection Γβ,N ∩ (Γβ,N + t) a self-similar set. We characterize the set in a geometrical and algebraical way, and give a sufficient and necessary condition for . Using techniques from beta-expansions, we show that there is a critical point βc ∊ (1/(2N − 1), 1/N), which is a transcendental number, such that has positive Hausdorff dimension if β ∊ (1/(2N − 1), βc), and contains countably infinite many elements if β ∊ (βc, 1/N). Moreover, there exists a second critical point such that has positive Hausdorff dimension if β ∊ (1/(2N − 1), αc), and contains countably infinite many elements if β ∊ [αc, 1/N).