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

Sándor, Csaba and Yang, Quan-Hui (2018) On some properties of representation functions related to the Erdős-Turan conjecture. EUROPEAN JOURNAL OF COMBINATORICS, 71. pp. 222-228. ISSN 0195-6698

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Abstract

For a set A\subsetN and n\in \mathbb{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\subset Z_m with 1 \le R_A(n) \le r for all n\in Z_m. In 2008, Chen proved that R_m \le 288 for all positive integers m. Recently the authors proved that R_m ≥ 6 for all integers m \ge 36. In this paper, for an abelian group G with |G| = m, we prove that if A\subset G satisfies R_A(g) \le 5 for all g\inG, then |{g : g\inG, R_A(g) = 0}| ≥ \frac{m}{4} − \sqrt{5m}. This improves a recent result of Li and Chen. We also give upper bounds of |{g : g\in G, R_A(g) = i}| for i = 2, 4.

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