Additive complements for two given asymptotic densities

Abstract

Let $0\le \alpha \le \beta \le 1$. For any finite set $B\subset \mathbb{N}$, we show that there exists a set $A\subset \mathbb{N}$ such that $\underline{d}(A+B)=\alpha$ and $\bar{d}(A+B)=\beta$, where $\underline{d}(A+B)$ and $\bar{d}(A+B)$ are the lower and upper asymptotic densities of the set $A+B$, respectively. This partially answers a question by Faisant et al. A theorem involving the so-called highly sparse sets was proved in the previous arXiv version of this note; however, as pointed out by Sai Teja Somu, the proof of the theorem was flawed. The theorem is now an open question.

arXiv comment4 pages. Remove Theorem 1.5 in the previous version