Sumsets with Prescribed Lower and Upper Asymptotic Densities

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.