Small sums of roots of unity

Alexandros Kalogirou

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arXiv · 2607.06098

Small sums of roots of unity

Alexandros Kalogirou

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Abstract

We address the question of how small a non-vanishing sum of $N$-th roots of unity with $k$ terms can be. We show upper bounds of the shape $N^{-\alpha_k}$, where $\alpha_k\rightarrow\infty$ with $k$. We also address the question of improving these bounds for a positive proportion of $N$.

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Alexandros Kalogirou. 2026-07-07. Small sums of roots of unity. https://arxiv.org/abs/2607.06098

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