2017/05/09 by Csaba Sándor, Sándor, Csaba, Quan-Hui Yang +1
Mathematics · Social Sciences · #11B13 #11B34 #China's Ethnic Minorities and Relations #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1705.03316
openalex publication_date 2017/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, we prove that if A⊆ ℤm satisfies RA(n)≤ 5 for all n∈ ℤm, then |\g:g∈ ℤm, RA(g)=0\|≥ (1)/(4)m-√(5m). This improves a recent result of Li and Chen. We also give upper bounds of |\g:g∈ ℤm, RA(g)=i\| for i=2,4.