Greedy Sidon sets for linear forms

The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in N that does not contain x(1), x(2), y(1), y(2) with x(1 )+ x(2) = y(1 )+ y(2). Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form x(1)+x(2) to arbitrary linear forms c(1)x(1 )+ & mldr; + c(h)x(h); these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when c(i) = n(i-1) for some n >= 2, and also when h = 2, c(1 )= 2, c(2 )>= 4, the "structured" domain. We also contrast the "enigmatic" domain when h = 2, c(1) = 2, c(2 )= 3 with the "structured" domain, and give upper bounds on the growth rates in both cases. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.