Some results on additive number theory

Some Results on Additive Number Theory'

Let 0 <a 1 <a2< . . . be any infinite sequence of integers . Denote by N(ai , n) the number of ai S n . I conjectured that to every sequence ai there corresponds a sequence b ; of density 0 (i .e ., such that lim n (1/n)N(b;, n)=0) so that every sufficiently large integer is of the form a i +b;. Lorentz 2 in a recent paper proved this conjecture ; in fact, he showed that there exists a sequence...

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 re...

Some Problems in Additive Number Theory

(3) f(x) = (log x/log 2) + 0(1)? 1\Mloser and I asked : Is it true that f(2 11) >_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) . P. Erdös, Problems and results in additive number theory, Colloque, Théorie des Nombres, Bruxelles 1955, p . 137 . 2. Let 1 < a 1< . . . < ak <_ x be a sequence of integers so that all the sums ai,+ . . .+ais, i 1 <...

Some Recent Developments in Additive Number Theory

Sidon in Additive Number Theory . on a Problem of Sidon in Additive Number Theory and on Some Related Problems Addendum

In a note in this Journal [16 (1941), 212-215], Turan and I proved, among other results, the following : Let a l < a2 < . . . < a, < n be a sequence of positive integers such that the sums aj+a; are all different . Then x < n'1 +0(n1 ) . On the other hand, there exist such sequences with x >n1(2---e), for any e >0 . Recently I noticed that J . Singer, in his paper "A theorem in finite projectiv...