在一个数有唯一展开的最小基数上
On the smallest base in which a number has a unique expansion
摘要：
给定一个实数$x>0$，我们确定$q_s(x):=\inf\mathscr{U}(x)$，其中$\mathscr{U}(x)$是所有基$q\in(1,2]$的集合，对于这些基，$x$具有$0$和$1$的唯一展开.我们给出了一个明确的描述$q_s(x)$的几个地区的$x$值。对于其他人，我们提出了一个有效的算法来确定$q_s(x)$和字典最小的唯一扩展的$x$。我们表明，下确界达到几乎所有的$x$，但也有一组点的正Hausdorff维数的下确界是适当的。此外，我们证明了函数$q_s$是右连续的，有左极限，没有向下的跳跃，并刻画了$q_s$的不连续点.本文的很大一部分致力于水平集$L(q):=\{x>0:q_s(x)=q\}$。我们表明，$L(q)$是有限的，几乎每一个$q$，但也有无穷多个无限水平集。特别是，对于Komornik-Loreti常数$q_{KL}=\min\mathscr{U}(1)\approx 1.787$，我们证明了$L(q_{KL})$有无穷多个左和无穷多个右聚点。
英文摘要：
Given a real number $x>0$, we determine $q_s(x):=\inf\mathscr{U}(x)$, where $\mathscr{U}(x)$ is the set of all bases $q\in(1,2]$ for which $x$ has a unique expansion of $0$'s and $1$'s. We give an explicit description of $q_s(x)$ for several regions of $x$-values. For others, we present an efficient algorithm to determine $q_s(x)$ 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 $q_s$ is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of $q_s$. A large part of the paper is devoted to the level sets $L(q):=\{x>0:q_s(x)=q\}$. 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 $q_{KL}=\min\mathscr{U}(1)\approx 1.787$ we prove that $L(q_{KL})$ has both infinitely many left- and infinitely many right accumulation points.
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