Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 1 / b / Solution

One Archimedean form of the Schmidt subspace theorem is as follows. Let $L_1,\dots ,L_n$ be linearly independent linear forms in $n$ variables with algebraic coefficients. For every $\epsilon >0$, all nonzero $\mathbf{x}\in {\mathbb{Z}}^n$ satisfyingbelong to a finite union of proper rational linear subspaces of ${\mathbb{Q}}^n$.

We will also use its finite-place form: if $S$ is a finite set of places containing the Archimedean ones and, for each $v\in S$, the forms $L_{1,v},\dots ,L_{n,v}$ are independent, then the integer solutions oflie in finitely many proper rational subspaces. The absolute values are normalized so that the product formula holds.