Scattered Sets and Roots of Unity in $\mathbb{Z}/p\mathbb{Z}$

TL;DR

This paper investigates the additive scattering properties of roots of unity in finite cyclic groups, revealing conditions under which these sets scatter or not, supported by extensive computational data for large primes.

Contribution

It characterizes when roots of unity in inite fields scatter under addition, providing theoretical results and extensive computational evidence for large primes.

Findings

Roots of unity do not scatter when 6 divides n.

They scatter for all but finitely many primes when 6 does not divide n.

Experimental data for primes up to 10^8 supports the theoretical results.

Abstract

If $\mathscr{G} = (G, +)$ is an abelian group, $S \subset G$ is said to scatter under addition if for all $a,b \in S$, $a+b \not \in S$. If $\mathscr{U}^{n}_{p}$ is the set of $n$th roots of unity in $\mathbb{Z}/p\mathbb{Z}$, where $n \geq 3$ is an integer and $p$ is a prime such that $n|(p-1)$, $\mathscr{U}^{n}_{p}$ does not scatter under addition when $6|n$, and $\mathscr{U}^{n}_{p}$ scatters under addition for all but a finite number of $p$ otherwise. Experimental data on the smallest, largest, and density of scattering modulus for $n \leq 10^8$ is also presented.