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

TL;DR

This paper investigates the sparsity of polynomial representations as sums of exponential functions, showing that such representations are rare among integers and establishing bounds on their frequency.

Contribution

It provides new bounds on the number of integers representable as sums of exponential functions, extending to more general inequalities involving polynomials and complex parameters.

Findings

Number of integers with polynomial exponential representations is negligible compared to rac{(\u2212)log M}{2}

Establishes bounds for representations involving complex polynomials and exponential parameters

Demonstrates sparsity of such representations among large integers

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).