Digital sum moments and substitutions-猫眼课题宝

数字总和矩和替换

DOI：

10.4064/aa-64-3-205-225

发表时间：

1993

期刊：

影响因子：

通讯作者：

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作者：

J. Dumont;Alain Thomas

k = 1时的结果首先由H. Delange（[De 75]），最近被推广到“正整数的θ-展开”的情况，θ是任意真实的基> 1（[GTi 91]），有一个o（1）修正项;在这种情况下，函数Fk，0被证明是连续的;对于二元展开，在k = 2的情况下，Fk，0和Fk，1也是连续的（见[C86]和[K90]）。本文的第一个目的是给出（1）的推广，包括在所谓的“与替换有关的计数系统”的框架内证明函数Fk，h的连续性。更精确地说，如果σ是有限字母表A上的一个本原替换，其最大特征值满足θ > 1，并且s（ν）表示∑ ni =1 f（mi）（其中对于整数ν，∑ ni =1| σ（mi）|是ν的唯一容许表示，f是从A到R的映射），然后证明了存在真实的数α和1-周期连续函数Fk，h（h = 0，1，. . .，k-1），使得对于任何x > 0，

The result in the case k = 1 was first established by H. Delange ([De75]) and has recently been generalized to the case of “θ-expansion of positive integers” with θ an arbitrary real base > 1 ([GTi91]), with an o(1) correction term; in this case the function Fk,0 is shown to be continuous; so are Fk,0 and Fk,1 in the case k = 2, for the binary expansion (see [C86] and [K90]). Our first aim in this paper is to give a generalization of (1), including the proof of the continuity of the functions Fk,h, within the framework of the so-called “numeration system associated with a substitution”. More precisely, if σ is a primitive substitution on a finite alphabet A whose largest eigenvalue satisfies θ > 1, and s(ν) denotes ∑n i=1 f(mi) (where for ν an integer, ∑n i=1 |σ(mi)| is the unique admissible representation of ν, and f is a map from A∗ to R), then we prove the existence of a real number α and 1-periodic continuous functions Fk,h (h = 0, 1, . . . , k−1) such that for any x > 0,

Digital sum moments and substitutions

数字总和矩和替换

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