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.
Categories: math.NT