Topology of univoque sets in real base expansions

Abstract

Given a positive integer M and a real number q∈(1,M+1], an expansion of a real number x∈[0,M/(q−1)] over the alphabet A={0,1,…,M} is a sequence (ci)∈AN such that x=∑i=1∞ciq−i. Generalizing many earlier results, we investigate in this paper the topological properties of the set Uq consisting of numbers x having a unique expansion of this form, and the combinatorial properties of the set Uq′ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [3] by adapting the method given in [12] for the case M=1.