Schmidt subspace theorem

Remarks

For $n = 2$, take $L_1(x, y) = \alpha y - x$ and $L_2(x, y) = y$ with $\alpha$ a real algebraic irrational. For $x/y$ close to $\alpha$, the norm $\norm{(x, y)}$ is at most a constant times $\abs{y}$. Up to this constant, the inequality then reads $\abs{\alpha - x/y} < \abs{y}^{-2-\varepsilon}$, and the theorem gives the Thue–Siegel–Roth theorem. The proof extends Roth's method with an auxiliary polynomial and uses the geometry of numbers. The conclusion speaks of finitely many subspaces, not finitely many points: in the example above, all the points $(x, 0)$ satisfy the inequality, because $L_2$ vanishes on them. Like Roth's theorem, the result is not effective.

Going deeper

For a student

The theorem is the main tool for finiteness results on Diophantine equations in several variables, such as $S$-unit equations $x_1 + \dots + x_n = 1$. It also gives transcendence results for numbers with special digit expansions. Versions with several places, the $p$-adic subspace theorem, are used for these applications.