On the sumsets of exceptional units in quaternion rings

TL;DR

This paper studies how elements in quaternion rings over finite rings can be expressed as sums of special units, providing formulas and bounds for the number of such representations depending on the ring's properties.

Contribution

It introduces a method to determine the number of representations of elements as sums of exceptional units in quaternion rings over local rings, including explicit counts and bounds.

Findings

For even order rings, explicit counts for sums of any number of exceptional units.

For odd order rings, bounds or exact counts for sums of two exceptional units.

Reduction to local rings simplifies the analysis of sumsets 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.