The additive structure of the squares inside rings

When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the sets underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in finite fields of prime order. Finally, for sufficiently small finite fields we computationally calculate the minimum for all prime orders.