On some properties of representation functions related to the Erdos-Turan conjecture
On some properties of representation functions related to the Erdos-Turan conjecture
复制标题
与鄂尔多斯-图兰猜想有关的表示函数的一些性质
DOI：
10.1016/j.ejc.2018.03.009
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发表时间：
2018
影响因子：
1
通讯作者：
Yang Quan-Hui
中科院分区：
文献类型：
--
作者：
S;or Csaba;Yang Quan-Hui
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. Recently the authors proved that R m≥ 6 for all integers m≥ 36. In this paper, for an abelian group G with| G|= m, we prove that if A⊆ G satisfies R A (g)≤ 5 for all g∈ G, then|{g: g∈ G, R A (g)= 0}|≥ 1 4 m− 5 m. This improves a recent result of Li and Chen. We also give upper bounds of|{g: g∈ G, R A (g)= i}| for i= 2, 4.
