On the density of \(B_2\)-bases (Q1071800)

\(A\subseteq\mathbb N\) is called a \(B_2\)-sequence if all differences \(a-a'\) for \(a,a'\in A\), \(a\neq a'\) are pairwise distinct. If, in addition, \(\{a-a'\mid a,a'\in A\}=\mathbb Z\) then \(A\) is called a \(B_2\)-basis. Let \(A(n):=| \{a\in A\mid a\leq n\}|\). P. Erdős [see \textit{H. Halberstam} and \textit{K. F. Roth}, Sequences (Reprint 1983; Zbl 0498.10001), p. 90] has shown that there are \(B_2\)-sequences \(A\) such that \(\lim \sup n^{-1/2} A(n)\geq 1/2\). \textit{M. Ajtai}, \textit{J. Komlós} and \textit{E. Szemerédi}, Eur. J. Comb. 2, 1--11 (1981; Zbl 0474.10038)] gave a random construction of a \(B_2\)-sequence \(A\) such that \(A(n)>(n \log n)^{1/3}/10^3\) for \(n>n_0\). In this note it is shown that these two results can also be obtained for \(B_2\)-bases by some modifications of the original proofs.