On some properties of representation functions related to the Erdős-Turán conjecture

For a set A⊆N and n∈N, let RA(n) denote the number of ordered pairs (a,a′)∈A×A such that a+a′=n. The celebrated Erdős-Turán conjecture says that, if RA(n)≥1 for all sufficiently large integers n, then the representation function RA(n) cannot be bounded. For any positive integer m, Ruzsa's number Rm is defined to be the least positive integer r such that there exists a set A⊆Zm with 1≤RA(n)≤r for all n∈Zm. In 2008, Chen proved that Rm≤288 for all positive integers m. Recently the authors proved that Rm≥6 for all integers m≥36. In this paper, for an abelian group G with |G|=m, we prove that if A⊆G satisfies RA(g)≤5 for all g∈G, then |{g:g∈G,RA(g)=0}|≥14m−5m. This improves a recent result of Li and Chen. We also give upper bounds of |{g:g∈G,RA(g)=i}| for i=2,4.