Intersections of homogeneous Cantor sets and beta-expansions

Let σß,N be the N-part homogeneous Cantor set with ß ε (1/(2N - 1), 1/N). Any string (jl)_∞ l _∞=1 with jl ε {0,±1, . . . ,±(N - 1)} such that t = l_∞ l=1 jlßl-1(1 - ß)/(N - 1) is called a code of t . Let Uß,±N be the set of t ε [-1, 1] having a unique code, and let Sß,±N be the set of t ε Uß,±N which makes the intersection σß,N n (σß,N + t) a self-similar set. We characterize the set Uß,±N in a geometrical and algebraical way, and give a sufficient and necessary condition for t ε Sß,±N. Using techniques from betaexpansions, we show that there is a critical point ßc ε (1/(2N - 1), 1/N), which is a transcendental number, such that Uß,±N has positive Hausdorff dimension if ß ε (1/(2N - 1), ßc), and contains countably infinite many elements if ß ε (ßc, 1/N). Moreover, there exists a second critical point ac = [N + 1 - v (N - 1)(N + 3)]/2 ε (1/(2N - 1), ßc) such that Sß,±N has positive Hausdorff dimension if ß ε (1/(2N -1), ac), and contains countably infinite many elements if ß ε [ac, 1/N).