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Additive Complements for a Given Asymptotic Density
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2021-01-07 , DOI: 10.1007/s00009-020-01679-0
Alain Faisant , Georges Grekos , Ram Krishna Pandey , Sai Teja Somu

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:

给定渐近密度的加法补码

我们调查子集的存在,\({\ mathbb {N}}:= \ {0,1,2,\点\} \) ,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\({\ mathbb {N}} \)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\(k \ ge 2。 \)

更新日期:2021-01-08
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