A lower bound of Ruzsa's number related to the Erdős-Turán conjecture

2016/12/27 by Sándor, Csaba, Yang, Quan-Hui
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1612.08722

Abstract

For a set A⊆ ℕ and n∈ ℕ, let R_A(n) denote the number of ordered pairs (a,a')∈ A× A such that a+a'=n. The celebrated Erdős-Turán conjecture says that, if R_A(n)≥ 1 for all sufficiently large integers n, then the representation function R_A(n) cannot be bounded. For any positive integer m, Ruzsa's number R_m is defined to be the least positive integer r such that there exists a set A⊆ ℤ_m with 1≤ R_A(n)≤ r for all n∈ ℤ_m. In 2008, Chen proved that R_m≤ 288 for all positive integers m. In this paper, we prove that R_m≥ 6 for all integers m≥ 36. We also determine all values of R_m when m≤ 35.

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