Topology of univoque sets in real base expansions

Abstract

Given a positive integer $M$ and a real number $q\in (1,M+1]$, an expansion of a real number $x\in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\dots ,M\}$ is a sequence $(c_i)\in A^{\mathbb{N}}$ such that $x={\sum }_{i=1}^{\infty }{c}_i{q}^{-i}$. Generalizing many earlier results, we investigate in this paper the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q^{\prime }$ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [B] by adapting the method given in [EJK] for the case $M=1$.

33 pages. arXiv admin note: text overlap with arXiv:math/0609708