Abstract
We show that for every natural number \(k\ge 2\) and real numbers \(\alpha _1,\alpha _2\), such that \(0\le \alpha _1 \le \alpha _2 \le 1\), there exists a subset A of \(\mathbb {N}=\{0,1,2,\ldots \}\), such that the lower and the upper asymptotic densities of \(kA=\{x_1+\cdots + x_k: x_i\in A,1\le i\le k\}\) are \(\alpha _1\) and \(\alpha _2\), respectively. We also show that there exists a set A, such that for every non-empty finite set of natural numbers B, the lower and the upper asymptotic densities of \(A+B=\{a+b : a\in A, b\in B\}\) are \(\alpha _1\) and \(\alpha _2\), respectively.
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Grekos, G., Pandey, R.K. & Somu, S.T. Sumsets with Prescribed Lower and Upper Asymptotic Densities. Mediterr. J. Math. 19, 201 (2022). https://doi.org/10.1007/s00009-022-02139-7
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DOI: https://doi.org/10.1007/s00009-022-02139-7