The units in a number ring are arithmetic gems that are poorly understood from the standpoint of arithmetic statistics. For example, what is the probability that the ring of integers of a number field contains a unit of mixed signature, i.e., a unit that has both a positive and a negative image in two distinct real embeddings? 

Absent theorems, we present Cohen-Lenstra style heuristics for unit signatures of odd abelian number fields.  In addition, we analyze the equation x^3 – ax^2 + bx – 1 = 0 and prove that there are infinitely many cyclic cubic fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.