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

Abstract

Given an integer g $g$ and also some given integers m $m$ (sufficiently large) and c 1 , ⋯ , c m $c_1,\dots , c_m$ , we show that the number of all non-negative integers n ⩽ M $n\leqslant M$ with the property that there exist non-negative integers k 1 , ⋯ , k m $k_1,\dots , k_m$ such that n 2 = ∑ i = 1 m c i g k i \begin{equation*} n^2=\sum _{i=1}^m c_i g^{k_i} \end{equation*} is o ( ( log M ) m − 1 / 2 ) $o((\log M)^{m-1/2})$ . We also obtain a similar bound when dealing with more general inequalities Q ( n ) − ∑ i = 1 m c i λ k i ⩽ B , \begin{equation*}\hskip1.8pc {\left|Q(n)-\sum _{i=1}^m c_i\lambda ^{k_i}\right|}\leqslant B,\hskip-2pc \end{equation*} where Q ∈ C [ X ] $Q\in {\mathbb {C}}[X]$ and also λ ∈ C $\lambda \in {\mathbb {C}}$ (while B $B$ is a real number).