Two infinite sets A,B of nonnegative integers are called additive complements,if all sufficiently large integers could be expressed as a+b,where a ∈ A,b ∈ B.Narkiewicz(1960)proved that:for any additive complements A,B,if limx→∞ A(x)B(x)/x=1,then limx→∞ A(2x)/A(x)=1,or this holds with A replaced by B.In this paper,we give a simplified proof for this result and generalize it.Furthermore,we also prove that:for any real numbers a,b with 1≤a≤2 and a≤b,there exists a set A of positive integers such that lim infx→∞ A(2x)/A(x)=a and lim supx→∞ A(2x)/A(x)=b.
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