On strong Sidon sets of integers

[摘要] A set S subset of N of positive integers is a Sidon set if the pairwise sums of its elements are all distinct, or, equivalently, if vertical bar(x + w) - (y + z)vertical bar >= 1 for every x, y, z, w epsilon S with x < y <= z < w. Let 0 <= alpha < 1 be given. A set S subset of N is an alpha-strong Sidon setif vertical bar(x + w) - (y + z)vertical bar >= w(alpha) for every x, y, z, w epsilon S with x < y <= z < w. 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 [n] = {1, . . . , n}, and give good bounds for F(n, alpha) = max vertical bar S vertical bar, where S ranges over all a-strong Sidon sets contained in[n]. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-10-01

[关键词] Sidon sets;Random sets of integers;Binary expansion