Independent Variables with Independent Sum and Difference: S1-Case

A classic result in probability theory states that two independent real-valued random variables having independent sum and difference are either constant or normally distributed with the same variance. In this article conditions are round on independent random variables X and Y taking values in the group of real numbers modulo 2[pi] so that X +Y and X - Y are independent. When X and Y are identically distributed, the small number of possible distributions for which X and Y have the desired property is known. In the general case there is a richer family of possible distributions for X and Y.