The almost sure number of pairwise sums for certain random integer subsets considered by P. Erdős (Q2759606)

Fix any \(\lambda> 0\). Let \(X_1,X_2,\dots\) be independent random variables taking only values zero and one as determined by the probabilities NEWLINE\[NEWLINEP\{X_j= 1\}= \min\Biggl\{\sqrt{{2\lambda\over \pi} {\ln j\over j}}, 1\Biggr\}.NEWLINE\]NEWLINE Let \({\mathcal G}_n= \sum^{[n/2]}_{j= 1} X_j X_{n-j}\). Then a.s. NEWLINE\[NEWLINE0\leq \liminf_{n\to\infty} {{\mathcal G}_n\over E{\mathcal G}_n}\equiv C_1(\lambda)< 1< C_2(\lambda)\equiv \limsup_{n\to\infty} {{\mathcal G}_n\over E{\mathcal G}_n}< \infty.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0971.00065].