On the smallest base in which a number has a unique expansion.

Given a real number x > 0, we determine qs(x) ≔ inf U(x), where U(x) is the set of all bases q∈ (1,2] for which x has a unique expansion of 0's and 1's. We give an explicit description of qs(x) for several regions of x-values. For others, we present an efficient algorithm to determine qs(x) and the lexicographically smallest unique expansion of x. We show that the infimum is attained for almost all x, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function qs is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of qs. A large part of the paper is devoted to the level sets L(q) ≔ x > 0 : qs(x) = q. We show that L(q) is finite for almost every q, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant qKL = min U(1) ≅ 1.787 we prove that L(qKL) has both infinitely many left- and infinitely many right accumulation points. [ABSTRACT FROM AUTHOR]