Note on a problem of Nathanson related to the $\varphi$-Sidon set

TL;DR

This paper proves that for any given set, there exists a polynomial-perturbed $ extit{Sidon}$ set of integers, addressing a recent problem posed by Nathanson and expanding understanding of $ extit{Sidon}$ sets in vector spaces.

Contribution

It demonstrates the existence of $ extit{Sidon}$ sets as polynomial perturbations of arbitrary sets, providing an affirmative answer to Nathanson's recent problem.

Findings

Existence of $ extit{Sidon}$ sets as polynomial perturbations of any set

Addresses a recent open problem by Nathanson

Provides new constructions in additive number theory

Abstract

Let $\varphi (x_{1}, \ldots, x_{h})=c_{1} x_{1}+\cdots+c_{h} x_{h}$ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $\varphi$-Sidon set if $\varphi\left(a_{1}, \ldots, a_{h}\right)=\varphi\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right)$ implies $\left(a_{1}, \ldots, a_{h}\right)=$ $\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right)$ for all $h$-tuples $\left(a_{1}, \ldots, a_{h}\right) \in A^{h}$ and $\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right) \in A^{h}$. We call $A$ a polynomial perturbation of $B$ if for some $r>0$ and positive integer $k_0$, $|a_k-b_k|< k^r$ holds for all integers $k \geq k_0$. In this paper, for a given set $B$, we prove that there exists a $\varphi$-Sidon set $A$ of integers that is a polynomial perturbation of $B$. This gives an affirmative answer to a recent problem of Nathanson. Some other results are also proved.