We investigate the existence of subsets A and B of \({\mathbb {N}}:=\{0,1,2,\dots \}\), such that the sumset \(A+B:=\{a+b:a\in A,b\in B\}\) has prescribed asymptotic density. We solve the particular case in which B is a given finite subset of \({\mathbb {N}}\) and also the case when \(B=A\); in the later case, we generalize our result to \(kA:=\{x_1+\cdots +x_k: x_i\in A, i=1,\dots ,k\}\) for an integer \(k\ge 2.\)

中文翻译：

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给定渐近密度的加法补码

我们调查子集的存在，甲和乙的\（{\ mathbb {N}}：= \ {0,1,2，\点\} \） ，使得sumset \（A + B：= \ {一个+ b：a \ a，b \ b \} \）具有规定的渐近密度。我们解决了B是\（{\ mathbb {N}} \）的给定有限子集的特殊情况，以及\（B = A \）的情况。在后一种情况下，我们将结果推广为\（kA：= \ {x_1 + \ cdots + x_k：x_i \ in A，i = 1，\ dots，k \} \）以获得整数\（k \ ge 2。 \）