Some Results on Additive Number Theory1

Let 0<ai<a2< ■ ■ -be any infinite sequence of integers. Denote by N(di, n) the number of a^n. I conjectured that to every sequence atthere corresponds a sequence bj of density 0 (i.e., such that lim„=0O (l/n)N(bj, n) = 0) so that every sufficiently large integer is of the form a,-+6yLorentz2 in a recent paper proved this conjecture; in fact, he showed that there exists a sequence bj with the required property satisfying for every n