Intersections of homogeneous Cantor sets and beta-expansions

Let Gamma(beta,N) be the N-part homogeneous Cantor set with beta is an element of (1/(2N - 1), 1/N). Any string (j(l))(l=1)(infinity) with j(l) is an element of {0, +/- 1, ... , +/- (N - 1)} such that t = Sigma(infinity)(l=1) j(l)beta(l) (1)(1 - beta)/(N - 1) is called a code of t. Let U-beta,U-+/- N be the set of t is an element of [-1, 1] having a unique code, and let S-beta,S-+/- N be the set of t is an element of U-beta,U-+/- N which makes the intersection Gamma(beta,N), boolean AND (Gamma(beta,N) + t) a self-similar set. We characterize the set U-beta,U-+/- N in a geometrical and algebraical way, and give a sufficient and necessary condition for t is an element of S-beta,S-+/- N. Using techniques from beta-expansions, we show that there is a critical point beta(c) is an element of (1/(2N - 1), 1/N), which is a transcendental number, such that U-beta,U-+/- N has positive Hausdorff dimension if beta is an element of (1/(2N - 1), beta(c)), and contains countably infinite many elements if beta is an element of (beta(c), 1/N). Moreover, there exists a second critical point alpha(c) = [N + 1 - root(N -1)(N + 3)]/2 is an element of (1/(2N - 1), beta(c)) such that S-beta,S-perpendicular to N has positive Hausdorff dimension if beta is an element of (1/(2N - 1), alpha(c)), and contains countably infinite many elements if beta is an element of [alpha(c), 1/N).