On small univoque bases of real numbers

Given a positive real number x , we consider the smallest base q s ( x ) ∈ ( 1 , 2 ) for which there exists a unique sequence ( d i ) of zeros and ones such that x = ∑ i = 1 ∞ d i ( q s ( x ) ) i . In this paper we give complete characterizations of those x ’s for which q s ( x ) ≤ q K L , where q K L is the Komornik–Loreti constant. Furthermore, we show that q s ( x ) = q K L if and only if x ∈ { 1 , q K L q K L 2 - 1 , 1 q K L 2 - 1 , 1 q K L ( q K L 2 - 1 ) } . Finally, we determine the explicit value of q s ( x ) if q s ( x ) < q K L .