Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2021-06-25 , DOI: 10.1016/j.jcta.2021.105490 Yoshiharu Kohayakawa , Sang June Lee , Carlos Gustavo Moreira , Vojtěch Rödl
A set of positive integers is a Sidon set if the pairwise sums of its elements are all distinct, or, equivalently, if for every with . Let be given. A set is an α-strong Sidon set if for every with . We prove that the existence of dense strong Sidon sets implies that randomly generated, infinite sets of integers contain dense Sidon sets. We derive the existence of dense strong Sidon sets from Ruzsa's well known result on dense Sidon sets [J. Number Theory 68 (1998), no. 1, 63–71]. We also consider an analogous definition of strong Sidon sets for sets S contained in , and give good bounds for , where S ranges over all α-strong Sidon sets contained in .
中文翻译:
关于强西顿整数集
一套 的正整数是一个西顿集,如果其元素的成对和都是不同的,或者等价地,如果 对于每个 和 . 让被给予。一套是一个α-强 Sidon 集,如果 对于每个 和 . 我们证明密集强西顿集的存在意味着随机生成的无限整数集包含密集西顿集。我们从 Ruzsa 在密集 Sidon 集上的众所周知的结果中推导出密集强 Sidon 集的存在 [J. 数论68 (1998),没有。1, 63–71]。我们还考虑了包含在中的集合S的强西顿集的类似定义,并给出很好的界限 其中小号范围在所有α -strong西顿集包含在.