Archiv der Mathematik ( IF 0.6 ) Pub Date : 2023-09-19 , DOI: 10.1007/s00013-023-01919-1 Shintaro Murakami , Yohei Tachiya
Let \((i,j)\in {\mathbb {N}}\times {\mathbb {N}}_{\ge 2}\) and \(S_{i,j}\) be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over \({\mathbb {Q}}\) of the numbers
$$\begin{aligned} 1, \quad \sum _{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in {\mathbb {N}}\times {\mathbb {N}}_{\ge 2}, \end{aligned}$$where \(b\ge 2\) is an integer and \(a_{i,j}(n)\) are bounded nonzero integer-valued functions on \(S_{i,j}\). Moreover, we also establish a necessary and sufficient condition on the subset \({\mathcal {A}}\) of \({\mathbb {N}}\times {\mathbb {N}}_{\ge 2}\) for the numbers
$$\begin{aligned} 1, \quad \sum _{n\in T_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in {\mathcal {A}}, \end{aligned}$$to be linearly independent over \({\mathbb {Q}}\) for any given infinite subsets \(T_{i,j}\) of positive integers. Our theorems generalize a result of V. Kumar.
中文翻译：
b 基数系统中某些数字的线性无关性
设\((i,j)\in {\mathbb {N}}\times {\mathbb {N}}_{\ge 2}\) 和 \(S_{i,j} \ )为正整数，包括某些算术级数中的所有素数。在本文中，我们证明了数字对\({\mathbb {Q}}\)的线性独立性
$$\begin{对齐} 1, \quad \sum _{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}} ,\quad (i,j)\in {\mathbb {N}}\times {\mathbb {N}}_{\ge 2}, \end{对齐}$$其中\(b\ge 2\)是一个整数，而\(a_{i,j}(n)\)是\(S_{i,j}\)上的有界非零整数值函数。此外，我们还对\({\mathbb {N}}\times {\mathbb {N}}_{\ge 2}\ )的子集\({\mathcal {A}}\)建立了充要条件）对于数字
$$\begin{对齐} 1, \quad \sum _{n\in T_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}} ,\quad (i,j)\in {\mathcal {A}}, \end{对齐}$$对于任何给定的正整数的无限子集\(T_{i,j}\)，与 \( {\mathbb {Q}} \)线性无关。我们的定理概括了 V. Kumar 的结果。
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