Greedy Sidon sets for linear forms

Abstract

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 $c_1{x}_1+\dots +c_h{x}_h$; 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 $n\ge 2$, and also when $h=2,c_1=2,c_2\ge 4$, 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.