Addition of sets of integers of positive density

Let A be a set of nonnegative integers, <(d)under bar(A)> its lower asymptotic density, and A + A = {a + a': a, a' is an element of A}. A classical theorem of Kneser completely describes the structure of all sets A subject to (d) under bar(A + A)<2 (d) under bar A. Freiman succeeded in partial generalization of Kneser's theorem when 2 is replaced by an arbitrary number. Going further in this direction, we obtain a mon precise version of Freiman's result. (C) 1997 Academic Press.