On the topological structure of univoque sets
On the topological structure of univoque sets
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DOI：
10.1016/j.jnt.2006.04.006
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发表时间：
2007
影响因子：
0.7
通讯作者：
V. Komornik;P. Loreti
中科院分区：
文献类型：
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作者：
V. Komornik;P. Loreti
őS，Horváth和Joó几年前发现，对于某些实数1<；q<；2，只存在一个由0和1组成的序列ciq，使得∑ciq−i=1。随后，这些数的集合U在[P.ErdőS，I.Joó，V.Komornik，唯一展开式的特征[公式]及相关问题，Bull，Bull]中被代数刻画。SoC。数学课。France 118(1990)377-390]和[V.Komornik，P.Loreti，次展开，超展开和非整数基唯一性性质，周期.数学课。亨加。44(2)(2002)195-216]。我们建立了U的闭包U‘的一个类似的刻画，这使得我们能够阐明这些集合的拓扑结构：U’∖U是U‘的可数稠密集，因此后者是完美的。此外，由于已知U有零勒贝格测度，U‘是一个Cantor集。
Erdős, Horváth and Joó discovered some years ago that for some real numbers 1<q<2 there exists only one sequence ciof zeroes and ones such that ∑ciq−i=1. Subsequently, the set U of these numbers was characterized algebraically in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions [Formula: see text] and related problems, Bull. Soc. Math. France 118 (1990) 377–390] and [V. Komornik, P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar. 44 (2) (2002) 195–216]. We establish an analogous characterization of the closure U¯ of U. This allows us to clarify the topological structure of these sets: U¯∖U is a countable dense set of U¯, so the latter set is perfect. Moreover, since U is known to have zero Lebesgue measure, U¯ is a Cantor set.