Linear independence of certain numbers in the base-b number system

Abstract

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.