  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.
