On the Distribution ofB3-Sequences

Abstract An infinite set of natural numbers is called aB3-sequence if all sumsa1+a2+a3withaj∈Aanda1⩽a2⩽a3are distinct. LetA(n) be the number of positive elements ⩽ninA. P. Erdos conjectures that everyB3-sequenceAsatisfies lim infn→∞ A(n) n−1/3=0. In this paper we prove that no sequence satisfyingA(n)∼αn1/3can be aB3-sequence. We also give other necessary conditions for aB3-sequence.

Some remarks on the Erdős-Turán conjecture