Additive complements for two given asymptotic densities

TL;DR

This paper demonstrates that for any finite set, one can construct an additive set with prescribed lower and upper asymptotic densities of their sum, partially answering a question in additive number theory.

Contribution

It establishes the existence of sets with prescribed asymptotic densities of their sum with any finite set, addressing a question by Faisant et al.

Findings

Existence of sets with specified lower and upper densities of A+B

Partial answer to a question by Faisant et al.

Open problem regarding highly sparse sets

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.