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

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 $\left|Q(n)-\sum_{i=1}^m c_iλ^{k_i}\right|\le B,$ where $Q\in {\mathbb C}[X]$ and also $λ\in {\mathbb C}$ (while $B$ is a real number).

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