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

Dragos Ghioca; Alina Ostafe; Sina Saleh; Igor E. Shparlinski

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arXiv · 2102.01949

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

Dragos Ghioca·Alina Ostafe·Sina Saleh·Igor E. Shparlinski

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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).

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BibTeX RIS

Dragos Ghioca, Alina Ostafe, Sina Saleh, Igor E. Shparlinski. 2021-02-03. On sparsity of representations of polynomials as linear combinations of exponential functions. https://arxiv.org/abs/2102.01949

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