Addition of Sets of Integers of Positive Density Original Research Article

LetAbe a set of nonnegative integers,imageAits lower asymptotic density, andA+A={a+a′ : a, a′set membership, variantA}. A classical theorem of Kneser completely describes the structure of all setsAsubject toimage(A+A)<2imageA. Freiman succeeded in partial generalization of Kneserʹs theorem when2is replaced by an arbitrary number. Going further in this direction, we obtain a more precise version of Freimanʹs result