On elements of sumsets with many prime factors (Q1801588)

Let \(\nu(n)\) be the number of distinct prime factors of \(n\). The following problem is studied in the paper. Having two finite sets of positive integers \({\mathcal A}\) and \({\mathcal B}\) how big is \(\nu(n)\) on the sumset \({\mathcal A}+{\mathcal B}\)? Suppose that \({\mathcal A}\) and \({\mathcal B}\) are subsets of \(\{n\leq N/2\}\). Then certainly \(\max\nu(a+b)\leq m\) where \(m=m(N)\) is the maximal value of \(\nu(n)\) for \(n\leq N\). It is shown that for dense sets this upper bound is almost attained, more precisely, for each \(\varepsilon>0\) there is a \(c(\varepsilon)\) such that if \(|{\mathcal A}| |{\mathcal B}|>\varepsilon N^2\) then we have \(\max\nu(a+b)>m- c(\varepsilon) \sqrt{m}\). It is also shown that this result is close to best possible. The proof has both probabilistic and combinatorial flavour.