P. Erdős, M. Horváth and I. Joó [Acta Math. Hung. 58, No. 3–4, 333–342 (1991; Zbl 0747.11005)] discovered continuum many real numbers \(1<q(<2)\) for which only one sequence \((c_i)=c_1 c_2\dots\) of integers \(c_i\in[0,q)\) satisfies the equality \(\sum_{i=1}^\infty c_i q^{-i}=1\). The set \(\mathcal U\) of such univoque numbers \(q>1\) has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.
In this paper the authors introduce for any fixed real number \(q>1\), the set \(\mathcal U_q\) of real numbers \(x\) for which exactly one sequence \((c_i)\) of integers \(c_i\in[0,q)\) satisfies the equality \(\sum_{i=1}^\infty c_i q^{-i}=x\). The purpose of this paper is to give a complete topological description of the sets \(\mathcal U_q\). The authors characterize their closures and determine those bases \(q\) for which \(\mathcal U_q\) is closed or even a Cantor set. They also study the set \(\mathcal U'_q\) consisting of all sequences \((c_i)\) of integers \(c_i\in[0,q)\) such that \(\sum_{i=1}^\infty c_i q^{-i}\in\mathcal U_q\). They determine the number \(r>1\) for which the map \(q\mapsto\mathcal U'_q\) (defined on \((1,\infty)\)) is constant in a neighborhood of \(r\) and the number \(q>1\) for which \(\mathcal U'_q\) is a subshift or a subshift of finite type.