论文标题
在数字具有独特扩展的最小基础上
On the smallest base in which a number has a unique expansion
论文作者
论文摘要
给定一个实际数字$ x> 0 $，我们确定$ q_s（x）：= \ inf \ mathscr {u}（x）$，其中$ \ mathscr {u}（x）$是所有基础$ q \ in（1,2] $的所有基础$ q \ in（1,2]的$ x $的独特扩展，以$ 0 $ $ $ $ $ $ q。 $ x $ - 对他人的价值，我们提出了一个有效的算法，以确定$ x $的词典，我们证明了$ x $的独特范围。跳跃，并表征$ q_s $的不连续点。 论文的很大一部分专门用于级别集合$ l（q）：= \ {x> 0：q_s（x）= q \} $。我们表明，几乎每$ Q $的$ L（Q）$都是有限的，但是也有无限的无限级别集。特别是，对于Komornik-loreti常数$ q_ {kl} = \ min \ mathscr {u}（1）\大约1.787 $，我们证明$ l（q_ {kl}）$具有无限的许多左和无限的右累积点。
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.
