Title:
Towards classifying dp-minimal expansions of the group of integers

Abstract:
Every known dp-minimal proper expansion of (Z,+) falls into one of three families: 
(1) (Z,+,<),
(2) reducts of dp-minimal expansions of (Z,+,C_\alpha), where C_\alpha is the cyclic order induced by the embedding n \mapsto n*alpha + Z of Z into R/Z, and 
(3) reducts of dp-minimal expansions of (Z,+,v), where v is a generalized valuation.
Are these really the only options, or are we yet to find examples of a completely new kind? 
In a previous work, I showed that the first family is characterized by an a-priori much weaker model-theoretic condition. I will present a new result, doing the same for the second family.