We prove a cell decomposition theorem for Presburger sets and
introduce a dimension theory for $Z$-groups with the Presburger
structure. Using the cell decomposition theorem we obtain a full
classification of Presburger sets up to definable bijection. We
also exhibit a tight connection between the definable sets in an
arbitrary p-minimal field and Presburger sets in its value group.
We give a negative result about expansions of Presburger
structures and prove uniform elimination of imaginaries for
Presburger structures within the Presburger language.