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

TL;DR

This paper proves the linear independence over rationals of certain infinite sums involving powers of a base and functions over specific integer subsets, generalizing previous results in number theory.

Contribution

It establishes new criteria for linear independence of these sums and generalizes a known result by V. Kumar.

Findings

Proves linear independence of sums over specific integer subsets.

Provides necessary and sufficient conditions for linear independence.

Generalizes previous results in the field of number theory.

Abstract

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.