Subexpansions,superexpansions and uniqueness properties in non-integer bases

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers 11. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section we answer a question concerning the existence of universal expansions, a notion introduced in [12]. This revised version was published online in June 2006 with corrections to the Cover Date.