Sidon sets for linear forms

Abstract

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