On sums with small prime factors
It is proved that every sufficiently large integer \(N>N_0(\epsilon)\) (where \(\epsilon>0\) is fixed) can be written in the form \(N=n_1+n_2+n_3\) such that all prime factors of \(n_1n_2n_3\) are bounded by \[ \text{exp}\left(\big(\sqrt{3/2}+\epsilon\big)\big(\log N\log\log N\big)^{1/2}\right). \] This is an improvement on a result of \textit{A. Balog} and \textit{A. Sárközy} [Stud. Sci. Math. Hung. 19, 35-47 (1984; Zbl 0569.10025)] only in the constant in the exp function. However, it is also shown that the above result does not leave too much room for further improvements, at least not with the present known approach. The main innovation of the paper is the use of the large sieve rather than the Hardy--Littlewood method.
scientific article; zbMATH DE number 868513