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

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.Comment: 9 page