On the sumsets of exceptional units in quaternion rings

Abstract

‎We investigate sums of exceptional units in a quaternion ring $H(R)$ over a finite commutative ring $R$‎. ‎We prove that in order to find the number of representations of an element in $H(R)$ as a sum of $k$ exceptional units for some integer $k \geq 2$‎, ‎we can limit ourselves to studying the quaternion rings over local rings‎. ‎For a local ring $R$ of even order‎, ‎we find the number of representations of an element of $H(R)$ as a sum of $k$ exceptional units for any integer $k \geq 2$‎. ‎For a local ring $R$ of odd order‎, ‎we find either the number or the bounds for the number of representations of an element of $H(R)$ as a sum of $2$ exceptional units‎.