Abstract
Let A and B be two subsets of the nonnegative integers. We call A and B additive complements if all sufficiently large integers n can be written as a+b, where a∈A and b∈B. Let S={1 2 ,2 2 ,3 2 ,···} be the set of all square numbers. Ben Green was interested in the additive complement of S. He asked whether there is an additive complement B={b n } n=1 ∞ ⊆ℕ which satisfies b n =π 2 16n 2 +o(n 2 ). Recently, Chen and Fang proved that if B is such an additive complement, then
Highlights
Two subsets A and B of nonnegative integers are said to be additive complements if their sum a + b (a ∈ A, b ∈ B)contains all sufficiently large integers
In [6], Erdos asked whether there exists a positive constant c such that l N > (1 + c)N
B is not an additive complement of S. From which they deduced that if B is an additive complement of S, lim sup n→∞
Summary
Two subsets A and B of nonnegative integers are said to be additive complements if their sum a + b (a ∈ A, b ∈ B)contains all sufficiently large integers. Fang proved that if B is such an additive complement, lim sup n→∞ They further conjectured that lim sup n→∞ We confirm this conjecture by giving a much more stronger result, i.e., lim sup π2 16 n Two subsets A and B of nonnegative integers are said to be additive complements if their sum a + b (a ∈ A, b ∈ B)
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