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arXiv · 2106.07808

Additive complements for two given asymptotic densities

Hung Viet Chu
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Abstract

Let $0\le α\le β\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = α$ and $\bar{d}(A+B) = β$, 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.

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Hung Viet Chu. 2022-07-03. Additive complements for two given asymptotic densities. https://arxiv.org/abs/2106.07808

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