2021/06/14 by Chu, Hung Viet
#11B05 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2106.07808
Let 0≤ α≤ β≤ 1. For any finite set B⊂ℕ, we show that there exists a set A⊂ℕ such that \underlined(A+B) = α and d(A+B) = β, where \underlined(A+ B) and 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.