Let \({\mathcal A}=(a_ m)\) and \({\mathcal B}=(b_ n)\) be two finite sequences of complex numbers. In what follows \({\mathcal A}\) and \({\mathcal B}\) are considered as characteristic functions of finite sets. Then \(S({\mathcal A},{\mathcal B})=\sum_{m-n=\ell^ 2}a_ mb_ n\ell\) provides information on the distribution of squares in the difference set \({\mathcal A}-{\mathcal B}\). Let \(\nu_ d(k)\) be the number of solutions to \(x^ 2\equiv k\pmod d\), and \(\chi_ c(k)=\sum_{d| c}\mu\left({c\over d}\right)\nu_ d(k)\) and \(s_ C(k)={1\over 2}\sum_{c\leq C}\chi_ c(k)\).
The following is the main theorem in the paper. Theorem. Let \(C\geq 2\) and \(a_ m=0\) for \(m>M\). Then \[ S({\mathcal A},{\mathcal B})=\sum_{m>n}a_ mb_ ns_ C(m-n)+{\mathcal E}_ C({\mathcal A},{\mathcal B}), \] where \({\mathcal E}_ C({\mathcal A},{\mathcal B})\ll(M/\sqrt C+M^{11/12}\log M)\|{\mathcal A}\| \|{\mathcal B}\|\) and \(\|{\mathcal A}\|=(\sum| a_ m|^ 2)^{1/2}\), and \(\|{\mathcal B}\|=(\sum| b_ n|^ 2)^{1/2}\). As an application the author shows that \[ \sum_{n<m\leq M}\Lambda(m)\Lambda(n)s(m-n)={1\over 4}M^ 2+O(M^{23/12}\log M). \]