On Cilleruelo-Nathanson's method in Sidon sets

Abstract

TextFor nonnegative integers h,g with h≥2, a set A of nonnegative integers is defined as a Bh[g] sequence if, for every nonnegative integer n, the number of representations of n with the form n=a1+a2+⋯+ah is no larger than g, where a1≤⋯≤ah and ai∈A for i=1,2,⋯,h. Let Z be the set of integers and N be the set of positive integers. In 2013, by introducing the method of Inserting Zeros Transformation, Cilleruelo and Nathanson obtained the following result: let f:Z→N⋃{0,∞} be any function such that lim inf|n|→∞f(n)≥g and let B be any Bh[g] sequence. Then, for any decreasing function ϵ(x)→0 as x→∞, there exists a sequence A of integers such that rA,h(n)=f(n) for all n∈Z and A(x)≫B(xϵ(x)). Recently, Nathanson further considered Sidon sets for linear forms. In this paper, we apply the Inserting Zeros Transformation into Sidon sets for linear forms and generalize the above result related to the inverse problem of representation functions. VideoFor a video summary of this paper, please visit https://youtu.be/KRtew2rLbXY.