On sparsity of representations of polynomials as linear combinations of exponential functions

Abstract

Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots ,c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots ,k_m$ such that $$n^2=\sum _{i=1}^m{c}_i{g}^{k_i}$$ is $o\left({\left(\log M\right)}^{m-1/2}\right)$. We also obtain a similar bound when dealing with more general inequalities $$\left|Q(n)-\sum _{i=1}^m{c}_i{\lambda }^{k_i}\right|\le B,$$ where $Q\in \mathbb{C}[X]$ and also $\lambda \in \mathbb{C}$ (while $B$ is a real number).