Positive-density sets A and B in the integers satisfy d(A+B)=d(A)+d(B) precisely when they are not both contained in a proper finite union of residue classes.
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An inverse theorem for sumsets of sets of positive density in the integers
Positive-density sets A and B in the integers satisfy d(A+B)=d(A)+d(B) precisely when they are not both contained in a proper finite union of residue classes.