  We consider the extension of two variable logic with quantifiers that state
that the number of elements where a formula holds should belong to a given
ultimately periodic set. We show that both satisfiability and finite
satisfiability of the logic are decidable. We also show that the spectrum of
any sentence is definable in Presburger arithmetic. In the process we present
several refinements to the ``biregular graph method''. In this method,
decidability issues concerning two-variable logics are reduced to questions
about Presburger definability of integer vectors associated with partitioned
graphs, where nodes in a partition satisfy certain constraints on their in- and
out-degrees.
