On small univoque bases of real numbers

Given a positive real number $x$, we consider the smallest base $q_s(x)\in(1,2)$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $$x=\sum_{i=1}^\infty\frac{d_i}{(q_s(x))^i}.$$ In this paper we give complete characterizations of those $x$'s for which $q_s(x)\le q_{KL}$, where $q_{KL}$ is the Komornik-Loreti constant. Furthermore, we show that $q_s(x)=q_{KL}$ if and only if $$x\in\left\{1, ~\frac{q_{KL}}{q_{KL}^2-1},~ \frac{1}{q_{KL}^2-1}, ~\frac{1}{q_{KL}(q_{KL}^2-1)}\right\}.$$ Finally, we determine the explicit value of $q_s(x)$ if $q_s(x)<q_{KL}$.