On the sumsets of exceptional units in $\mathbb{Z}_n$

Let $R$ be a commutative ring with $1\in R$ and $R^{\ast}$ be the multiplicative group of its units. In 1969, Nagell introduced the exceptional unit $u$ if both $u$ and $1-u$ belong to $R^{\ast}$. Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$. In this paper, given an integer $k\ge 2$, we obtain an exact formula for the number of ways to represent each element of $\mathbb{Z}_n$ as the sum of $k$ exceptional units. This generalizes a recent result of J. W. Sander for the case $k=2$.