On the sumsets of units and exceptional units in residue class rings

Abstract

Let $n\ge 1,e\ge 1,t,k\ge 2$ and c be integers such that $0\le t\le k$. An integer u is called a unit in the ring $Z_n$ of residue classes modulo n if $\gcd (u,n)=1$. Let $Z_n^{⁎}$ be the multiplicative group of $Z_n$. A unit u is called an exceptional unit in the ring $Z_n$ if $1-u\in Z_n^{⁎}$. We write $Z_n^{⁎⁎}$ for the set of all exceptional units of $Z_n$. We denote by $M_{t,k,c,e}\left(n\right)$ the number of representations of the element $c\in Z_n$ as the sum of e-th powers of t units and e-th powers of $k-t$ exceptional units in the ring $Z_n$. When $t=k$, Brauer determined the number $M_{k,k,c,1}\left(n\right)$ which answers a question of Rademacher. Mollahajiaghaei gave a formula for $M_{k,k,c,2}\left(n\right)$. When $t=0$, Sander presented a formula for $M_{0,2,c,1}\left(n\right)$, and later on Yang and Zhao got an exact formula for $M_{0,k,c,1}\left(n\right)$. However, when $1\le t\le k-1$, the formulas for the numbers $M_{t,k,c,1}\left(n\right)$ and $M_{t,k,c,2}\left(n\right)$ are unknown so far. In this paper, we answer this question. In fact, by using algebraic method, Hensel's lemma and exponential sums, we deduce explicit formulas for the numbers $M_{t,k,c,1}\left(n\right)$ and $M_{t,k,c,2}\left(n\right)$ for any integer t with $0\le t\le k$. From this, we derive an exact formula for the number of representations of an element c as the sum of a unit and an exceptional unit in $Z_n$. We also obtain an explicit formula for the number of representations of an element c as the sum of square of a unit and square of an exceptional unit in $Z_n$. Our result gives a uniform generalization to the theorems of Brauer, of Sander, of Yang and Tang, of Mollahajiaghaei, of Yang and Zhao, and of Feng and Hong.