Inverse results for restricted sumsets in Z/pZ

Let p be a prime, A and B be subsets of Z/pZ and S be a subset of AxB. We write A(S)+B:= {a+b:(a,b)is an element of S}. In the first inverse result of this paper, we show that if divided by A(S)+B divided by divided by and| (AxB)S| are small, then A has a big subset with small difference set. In the second theorem of this paper, we use the previous result to show that if divided by AS+B divided by, |A| and |B| ares mall, then big parts of A and B are contained in short arithmetic progressions with the same difference. As an application of this result, we get an inverse of Pollard's theorem