Small sums of roots of unity

Alexandros Kalogirou

Small sums of roots of unity

Number Theory 2026-07-07 v1

Authors: Alexandros Kalogirou

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 $N$ .

Keywords

roots of unity vanishing sums exponential sums upper bounds

Cite

@article{arxiv.2607.06098,
  title  = {Small sums of roots of unity},
  author = {Alexandros Kalogirou},
  journal= {arXiv preprint arXiv:2607.06098},
  year   = {2026}
}