Sums of exceptional units in residue class rings

Given a commutative ring R   with 1∈R1∈R and the multiplicative group R⁎R⁎ of units, an element u∈R⁎u∈R⁎ is called an exceptional unit   if 1−u∈R⁎1−u∈R⁎, i.e., if there is a u′∈R⁎u′∈R⁎ such that u+u′=1u+u′=1. We study the case R=Zn:=Z/nZR=Zn:=Z/nZ of residue classes modn and determine the number of representations of an arbitrary element c∈Znc∈Zn as the sum of two exceptional units. As a consequence, we obtain the sumset Zn⁎⁎+Zn⁎⁎ for all positive integers n  , with Zn⁎⁎ denoting the set of exceptional units of ZnZn.