We prove that there exists an absolute constant c > 0 such that for any finite integer set A with |A| > 1 and any positive integer m < c|A|/ ln |A| there are at most m positive integers b satisfying |(A + b) \ A| ≤ m; equivalently, there are at most m positive integers possessing |A| − m (or more) representations as a difference of two elements of A. This is best possible in the sense that for every positive integer m there exists a finite integer set A with |A| > m log2 (m/2) such that |(A + b) \ A| ≤ m holds for b = 1, ..., m + 1. Joint work with Sergei Konyagin.