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

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.