Inverse results for restricted sumsets in $${\mathbb {Z}/p\mathbb {Z}}$$

Let p be a prime, A and B be subsets of \({\mathbb {Z}/p\mathbb {Z}}\) and S be a subset of \(A\times B\). We write \(A{{\mathop {+}\limits ^{S}}}B:=\{a+b:\;(a,b)\in S\}\). In the first inverse result of this paper, we show that if \(\left| A{{\mathop {+}\limits ^{S}}}B\right| \) and \(|(A\times B)\setminus S|\) 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 \(\left| A{{\mathop {+}\limits ^{S}}}B\right| \), |A| and |B| are small, 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.