  Given a binary form $F \in \mathbb{Z}[X, Y]$, we define its value set to be
$\{F(x, y) : (x, y) \in \mathbb{Z}^2\}$. Let $F, G \in \mathbb{Z}[X, Y]$ be two
binary forms of degree $d \geq 3$ and with non-zero discriminant. In a series
of three papers, we will give necessary and sufficient conditions on $F$ and
$G$ to have the same value set. These conditions will be entirely in terms of
the automorphism groups of the forms.
  In this paper, we will build the general theory that reduces the problem to a
question about lattice coverings of $\mathbb{Z}^2$, and we solve this problem
when $F$ and $G$ have a small automorphism group. The larger automorphism
groups $D_4$ and $D_3, D_6$ will respectively be treated in part II and part
III.
