Difference Sets and Algebraic Number Theory an Exposition of Some Results of Schmidt

Several combinatorical problems reduce to classifying certain families of difference sets. Schmidt, in [1], obtained strong new nonexistence results for this class of problem by applying algebraic number theory: a difference set can be related to a cyclotomic integer with bounded coefficients and a prescribed absolute value, and in many cases these integers can be number-theoretically proven not to exist. We lay out the core of Schmidt’s results, simplifying what we can and omitting some of the less interesting details.