Sharp threshold asymptotics for the emergence of additive bases (Q2855588)

The paper is devoted to the random additive bases. We select each integer \(m\in\{0,1,2,\ldots,n\}\) independently with some probability \(p_n\) depen\-ding only on \(n\). In such a way, we obtain a random set \(\mathcal{A}\). Let \(\alpha\in(0,1)\) and \(X\) be the number of integers in the interval \([\alpha n, (2-\alpha)n]\) that cannot be written in the form \(x_1+x_2+\ldots +x_k,\;x_1,\ldots,x_k\in\mathcal{A}\). We observe that \(X=0\) if and only if \(\mathcal{A}\) is a truncated additive \(k\)-basis, i.e. \(m=x_1+x_2+\ldots +x_k\), \ \( x_1,\ldots,x_k\in\mathcal{A}\) for each \(m\in[\alpha n, (2-\alpha)n]\). In the paper, the cases when \(k=2\) and \(k\ge 2\) are considered separately. The following statement is typical.NEWLINENEWLINESuppose that \(k=2\), and NEWLINE\[CARRIAGE_RETURNNEWLINEp_n=\sqrt{((2\log n-2\log\log n)/\alpha+A_n)/n}CARRIAGE_RETURNNEWLINE\]NEWLINE with some suitable sequence \(A_n\). When \(n\rightarrow\infty\), there are three distinct cases:NEWLINENEWLINE(i) if \(A_n\rightarrow\infty\) then \(\mathbb{P}(X=0)\rightarrow 1\);NEWLINENEWLINE(ii) if \(A_n\rightarrow -\infty\) then \(\mathbb{P}(X=0)\rightarrow 0\);NEWLINENEWLINE(iii) if \(A_n\rightarrow A\) then \(\mathbb{P}(X=0)\rightarrow\exp\{-2\alpha\exp\{-\alpha A/2\}\}\).