How often can a given distance occur in a finite set of integers?

Let n, a, d be natural numbers and A a set of integers of the closed interval [0, n] with | A | = a. Then we establish sharp lower and upper bounds for the number of pairs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $(x,y)\in A\times A$\end{document} for which y–x = d. Roughly spoken, we investigate how often a distance d can occur in A.