论文标题
基础中某些数字的线性独立性-B $编号系统
Linear independence of certain numbers in the base-$b$ number system
论文作者
论文摘要
令$（i，j）\ in \ mathbb {n} \ times \ mathbb {n} _ {\ geq2} $和$ s_ {i，j} $是一个无限的积极整数子集，包括所有算术进步中所有质量数字。在本文中，我们证明了数字的$ \ mathbb {q} $的线性独立性\ [1，\ quad \ sum_ {n \ in s_ {i，j}}}}}^{} \ frac {a__ {a_ {a_ {i，j} \ Mathbb {n} \ times \ Mathbb {n} _ {\ geq2}，\]，其中$ b \ geq2 $是整数，而$ a_ {i，j}（n）$是限制在$ s_ {i，j} $上的$ seroger integer noger integer bote。此外，我们还在子集$ \ m rathcal {a} $ of $ \ mathbb {n} \ times \ times \ mathbb {n} _ {\ geq2} $ for numbers \ [1，\ quad \ sum_ {n \ in {n \ in { t_ {i，j}}^{} \ frac {a_ {a_ {i，j}（n）} {b^{b^{in^j}}，\ quad（i，j）\ in \ mathcal {A a} \ in \ mathcal {a} \]与任何$ \ \ m raster in in ost of $ \ mathbb {q MathBb {整数。我们的定理概括了V. Kumar的结果。
Let $(i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}$ 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 \[ 1, \quad \sum_{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}, \] where $b\geq2$ 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 $\mathbb{N}\times \mathbb{N}_{\geq2}$ for the numbers \[ 1, \quad \sum_{n\in T_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in \mathcal{A} \] 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.
