  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, for all
$h$-tuples $(a_1,\ldots, a_h) \in A^h$ and $ (a'_1,\ldots, a'_h) \in A^h$, the
relation $\varphi(a_1,\ldots, a_h) = \varphi(a'_1,\ldots, a'_h)$ implies
$(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. There exist infinite Sidon sets for
the linear form $\varphi$ if and only if the set of coefficients of $\varphi$
has distinct subset sums. In a normed vector space with $\varphi$-Sidon sets,
every infinite sequence of vectors is asymptotic to a $\varphi$-Sidon set of
vectors. Results on $p$-adic perturbations of $\varphi$-Sidon sets of integers
and bounds on the growth of $\varphi$-Sidon sets of integers are also obtained.
