Almost all binary forms of degree $\ge 3$ fail to represent a fixed integer

We prove that for any fixed integer $n\ge 3$ and nonzero integer $m$, the proportion of integral binary forms of degree $n$ that represent $m$ tends to zero as the height tends to infinity. In fact, almost all such forms fail to represent $m$. Our method uses lattice point counting and geometric methods, including Davenport’s lemma and estimates for volumes of hyperplane sections of cubes, together with an analysis of the distribution of rational points on such hyperplanes. The result also holds when restricted to irreducible forms.