On the characterization of canonical number systems
On the characterization of canonical number systems
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关于规范数系统的表征
DOI：
10.18910/6910
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发表时间：
2004
影响因子：
0.4
通讯作者：
J. Thuswaldner
中科院分区：
文献类型：
--
作者：
K. Scheicher;J. Thuswaldner
众所周知，每个正整数 n 都可以唯一地表示为和 n = d0 + d1b + 。 。 。 + dhb h ，其整数基数 b ≥ 2，dh 6= 0 且 di ∈ {0, . 。 。 , b − 1}。这个概念可以概括为几个方向。一方面，基本序列 1, b, b, . 。 。可以用序列 1 = u0 < u1 < u2 < 替换。 。 。获得正整数的表示。特别令人感兴趣的是序列 {ui}i=0 由线性递推定义的情况。属于此类的一个著名例子是所谓的 Zeckendorf 表示。另一方面，我们可以概括可以表示的一组数字。我们提到属于此类的两种数系： Rényi [27] 引入的所谓 β 展开，它是将单位区间内的实数表示为实数基数 β 的幂和。如果 β 是线性循环基本序列 {ui}i=0 的特征多项式的零，则实数的这些数字表示与正整数的数字表示密切相关。特别有趣的是 β 是皮索数的情况。这些扩展已被广泛研究。我们在此提到论文 Berend-Frougny [6]、Frougny [12, 13]、Frougny-Solomyak [14, 15] 和 Loraud [25] 并参考其中给出的参考文献。另一种允许与 N 不同的集合表示的数系是所谓的规范数系（简称 CNS）。由于中枢神经系统构成了本文研究的主要对象，我们回顾一下它们的定义（参见 Akiyama-Pethő [2]）。
It is well known that each positive integer n can be expressed uniquely as a sum n = d0 + d1b + . . . + dhb h with an integral base number b ≥ 2, dh 6= 0 and di ∈ {0, . . . , b − 1}. This concept can be generalized in several directions. On the one hand the base sequence 1, b, b, . . . can be replaced by a sequence 1 = u0 < u1 < u2 < . . . to obtain representations of positive integers. Of special interest is the case where the sequence {ui}i=0 is defined by a linear recurrence. A famous example belonging to this class is the so-called Zeckendorf representation. On the other hand, one can generalize the set of numbers which can be represented. We mention two kinds of number systems belonging to this class: The so called β-expansions introduced by Rényi [27] which are representations of real numbers in the unit interval as sums of powers of a real base number β. These digit representations of real numbers are strongly related to digit representations of positive integers if β is a zero of the characteristic polynomial of a linear recurring base sequence {ui}i=0. Of special interest is the case where β is a Pisot number. These expansions have been extensively studied. We mention here the papers Berend-Frougny [6] , Frougny [12, 13], Frougny-Solomyak [14, 15] and Loraud [25] and refer to the references given there. Another kind of number systems which admit the representation of a set which is different from N are the so-called canonical number systems (for short CNS). Since CNS form the main object studied in the present paper we recall their definition (cf. Akiyama-Pethő [2]).