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

Given a real number x &gt; 0 x&gt;0 , we determine alttext="q Subscript s Baseline left-parenthesis x right-parenthesis colon-equal inf script upper U right-parenthesis"> q s ( stretchy="false">) ? inf ? U encoding="application/x-tex">q_s(x)?\operatorname {inf}{\mathscr {U}}(x) where alttext="script encoding="application/x-tex">{\mathscr is the set of all bases element-of 1 comma 2 right-bracket"> ?<!-- ? <mml:mn>1 , 2 stretchy="false">] encoding="application/x-tex">q\in (1,2] for which alttext="x"> encoding="application/x-tex">x has unique expansion alttext="0"> encoding="application/x-tex">0 ’s and alttext="1"> encoding="application/x-tex">1 ’s. We give an explicit description encoding="application/x-tex">q_s(x) several regions -values. For others, present efficient algorithm to lexicographically smallest . show that infimum attained almost but there also points positive Hausdorff dimension proper. In addition, function s"> encoding="application/x-tex">q_s right-continuous with left-hand limits no downward jumps, characterize discontinuity A large part paper devoted level sets alttext="upper L q StartSet 0 colon equals EndSet"> L fence="false" stretchy="false">{ : = stretchy="false">} encoding="application/x-tex">L(q)?\{x&gt;0:q_s(x)=q\} encoding="application/x-tex">L(q) finite every alttext="q"> encoding="application/x-tex">q are infinitely many infinite sets. particular, Komornik-Loreti constant K min almost-equals 1.787"> K min ?<!-- ? <mml:mn>1.787 encoding="application/x-tex">q_{KL}=\operatorname {min}{\mathscr {U}}(1)\approx 1.787 prove encoding="application/x-tex">L(q_{KL}) both left- right accumulation points.