Unique expansions of real numbers
Unique expansions of real numbers
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DOI：
10.1016/j.aim.2008.12.008
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发表时间：
2006-09
影响因子：
1.7
通讯作者：
M. D. Vries;V. Komornik
中科院分区：
文献类型：
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作者：
M. D. Vries;V. Komornik
几年前发现，存在非整数真实的数q>1，其中只有一个整数序列ci∈[0，q）满足等式∑i=1∞ciq−i=1。这类“univoque数”的集合具有丰富的拓扑结构，它的研究揭示了一些意想不到的联系与测度论，分形，遍历理论和丢番图近似。在本文中，我们考虑对于每个固定的q>1，真实的数x的集合Uq具有形式为∑i=1∞ciq−i=x的唯一表示，其中整数ci属于[0，q）。我们进行了详细的拓扑研究，这些集。例如，我们刻画了它们的闭包，我们确定了Uq闭的基q，甚至是康托集。我们还研究了由整数ci∈[0，q）的所有序列（ci）组成的集合Uq′，使得∑i=1∞ciq−i∈Uq.我们确定了在r的邻域中映射q <$Uq′（定义在（1，∞）上）为常数的数r>1和Uq′为子移位或有限型子移位的数q>1。
It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality ∑i=1∞ciq−i=1. The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed q>1 the set Uqof real numbers x having a unique representation of the form ∑i=1∞ciq−i=x with integers cibelonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uqis closed or even a Cantor set. We also study the set Uq′consisting of all sequences (ci) of integers ci∈[0,q) such that ∑i=1∞ciq−i∈Uq. We determine the numbers r>1 for which the map q↦Uq′(defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which Uq′is a subshift or a subshift of finite type.