The analogue of Erdős–Turán conjecture in Zm Original Research Article

Given a set Asubset ofN let σA(n) denote the number of ordered pairs (a,a′)set membership, variantA×A such that a+a′=n. The celebrated Erdős–Turán conjecture states that if Asubset ofN such that σA(n)greater-or-equal, slanted1 for all sufficiently large n, then the representation function σA(n) must be unbounded. For each positive integer m, let Rm be the least positive integer r such that there exists a set Asubset of or equal toZm with A+A=Zm and σA(n)less-than-or-equals, slantr. Ruzsaʹs method in [I.Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990) 145–151] implies that Rm must be bounded. It is pleasure to call Rm a Ruzsaʹs number. In this paper we prove that all Ruzsaʹs numbers Rmless-than-or-equals, slant288. This improves the previous bound Rmless-than-or-equals, slant5120. Several related open problems are proposed. Video abstract For a video summary of this paper, please visit http://www.youtube.com/watch?v=hgDwkwg_LzY.