The analogue of Erdős–Turán conjecture in ZmZm

Given a set A⊂NA⊂N let σA(n)σA(n) denote the number of ordered pairs (a,a′)∈A×A(a,a′)∈A×A such that a+a′=na+a′=n. The celebrated Erdős–Turán conjecture states that if A⊂NA⊂N such that σA(n)⩾1σA(n)⩾1 for all sufficiently large n  , then the representation function σA(n)σA(n) must be unbounded.For each positive integer m  , let RmRm be the least positive integer r   such that there exists a set A⊆ZmA⊆Zm with A+A=ZmA+A=Zm and σA(n)⩽rσA(n)⩽r. Ruzsa's method in [I.Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990) 145–151] implies that RmRm must be bounded. It is pleasure to call RmRm a Ruzsa's number  . In this paper we prove that all Ruzsa's numbers Rm⩽288Rm⩽288. This improves the previous bound Rm⩽5120Rm⩽5120. Several related open problems are proposed.Video abstractFor a video summary of this paper, please visit http://www.youtube.com/watch?v=hgDwkwg_LzY.