Cyclotomy over products of finite fields and combinatorial applications

We introduce a new kind of cyclotomy over a cartesian product RR of finitely many finite fields, which generalizes the classical cases of only one or two fields. We describe the orbits corresponding to this cyclotomy and, to a great extent, we determine the arithmetic corresponding to these orbits in the group ring ZRZR, i.e. given three orbits AA, BB and CC, we study how many ways there are of expressing an element of CC as a sum of two elements of AA and BB. In particular, we obtain the cyclotomic numbers in a variety of interesting cases. We exhibit some applications of this cyclotomy to the construction of combinatorial structures with nice groups of multipliers. More precisely, we produce an infinite family of divisible difference sets with new parameters, another family of relative difference sets, as well as some infinite families and some sporadic examples of partial difference sets. We also obtain both infinite families and a sporadic construction of three-class association schemes.