Sidon sets for linear forms

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