Subexpansions, superexpansions and uniqueness properties in non-integer bases

Abstract

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Reenyi and then investigated by many authors. Some years ago, Erdős, Horvath and Joo found the surprising fact that there exist infinitely many numbers 1 1. 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].