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

Abstract

Let $\phi (x_1,\dots ,x_h)=c_1{x}_1+\cdots +c_h{x}_h$ 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 $\phi$-Sidon set if $\phi \left(a_1,\dots ,a_h\right)=\phi \left(a_1^{\prime },\dots ,a_h^{\prime }\right)$ implies $\left(a_1,\dots ,a_h\right)=$ $\left(a_1^{\prime },\dots ,a_h^{\prime }\right)$ for all $h$-tuples $\left(a_1,\dots ,a_h\right)\in A^h$ and $\left(a_1^{\prime },\dots ,a_h^{\prime }\right)\in A^h$. 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 $k\ge k_0$. In this paper, for a given set $B$, we prove that there exists a $\phi$-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.

6 pages. arXiv admin note: text overlap with arXiv:2101.01034

References in corpus (2)

• The Bose-Chowla argument for Sidon sets
• Sidon sets for linear forms