UNIQUE REPRESENTATIONS OF REAL NUMBERS IN NON-INTEGER BASES
UNIQUE REPRESENTATIONS OF REAL NUMBERS IN NON-INTEGER BASES
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DOI：
10.4310/mrl.2001.v8.n4.a12
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发表时间：
2001-07
影响因子：
1
通讯作者：
P. Glendinning;N. Sidorov
中科院分区：
文献类型：
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作者：
P. Glendinning;N. Sidorov
从 Renyi [16] 和 Parry [15] 的开创性著作开始，自 20 世纪 50 年代末以来，与非整数基实数展开相关的问题就得到了系统的研究。最初的方法基于选择“数字”的特定算法（例如贪婪扩展）。这通常会产生所讨论的所有可能实数（例如，非负数或属于给定区间）的数字序列集，与经典的 d-adic 情况不同，它不是笛卡尔积，而是具有复杂的结构。然而，在 1990 年代，由 Paul Erdos 领导的一群匈牙利数学家开始研究提供实数独特表示的 0-1 序列 [6,7,8]。本文继续这一研究方向。我们的设置如下。令 q ∈ (1, 2) 为我们的参数，并且 Σ = ∏∞1 {0, 1}；我们考虑那些在 q 的基础上具有独特扩展形式的 x
Problems related to the expansions of real numbers in non-integer bases have been systematically studied since the late 1950’s, starting with the seminal works by Renyi [16] and Parry [15]. The original approach is based on a specific algorithm for choosing “digits” (e.g. the greedy expansions). This usually leads to the set of sequences of digits for all possible real numbers in question (for instance, non-negative or belonging to a given interval) which, unlike the classical d-adic case, is not a Cartesian product but has a complicated structure. However, in the 1990’s a group of Hungarian mathematicians led by Paul Erdos began to investigate 0-1 sequences that provide unique representations of reals [6, 7, 8]. The present paper continues this line of research. Our set-up is as follows. Let q ∈ (1, 2) be our parameter and Σ = ∏∞1 {0, 1}; we consider those x which have unique expansions in base q of the form