Pairs of sets with small sumset and small periodic product-set (Q765178)

We denote by \(\text{card}(A)\) the number of elements of a finite set \(A\). The main result of the paper is as follows. Given two finite subsets \(A,B\) of a field \(\mathbb{F}\) of characteristic \(r\) containing both at least one element; we have \(\text{card}(AB) = \max(\text{card}(A),\text{card}(B))\) and \(\text{card}(A+B) = \min(p,\text{card}(A)+\text{card}(B)-1)\) if and only if any of ten explicit conditions hold. Here \(r=p\) when \(r>0\) and \(p=\) infinity otherwise. For example, one of the conditions is: \(A=d \{0,1\}\) and \(B\) is an arithmetic progression with difference \(d\) that contains \(0\), for some nonzero \(d \in \mathbb{F}.\) Polynomials that have roots in arithmetic or geometric progressions are explicit determined as tools in the proof, as well as classical results as Kneser's theorem are skillfully applied.