Constructions of Sequences of Alternating Sum and Difference Dominated Sets

A More Sums Than Difference (MSTD) set is a finite set of integers $A$ where the cardinality of its sumset, $A+A$, is greater than the cardinality of its difference set, $A-A$. Since addition is commutative while subtraction isn't, it was conjectured that MSTD sets are rare. As Martin and O'Bryant proved a small (but positive) percentage are MSTD, it is natural to ask what additional properties can we impose on a chain of MSTD sets; in particular, can we construct a sequence of sets alternating between being MSTD and More Difference Than Sums (MDTS) where each properly contains the previous? We provide several such constructions; the first are trivial and proceed by filling in all missing elements from the minimum to maximum elements of $A$, while the last is a more involved construction that prohibits adding any such elements.