Quotients of dense subsets of integers and short distances between product elements (Q2113407)

By using results and methods proposed by \textit{J. Cilleruelo} et al. [Bull. Lond. Math. Soc. 42, No. 3, 517--526 (2010; Zbl 1205.11016)], the author refines a lower bound on the set of (Minkowski) quotients of two subsets \(A\) and \(B\) from finite intervals of natural numbers with positive density. As a consequence, he proves that if \(a\) and \(b\) are real numbers from \((0,1]\) , and if \(A\) and \(B\) are infinite subsets of natural numbers whose upper asymptotic densities are \(a, b\) respectively , and by putting \(d=2/ab\) , then there exist infinitely many pairs \((u,v)\in AB\) , \(u\ne v\) such that \(\lvert u-v\rvert\) =O(exp((c.log d/log log d)/ab)), c being an absolute constant, and \(AB\) denoting the Minkowski product of the sets \(A\) and \(B\).