---
tag: 9TPS3W497D
kind: statement
role: theorem
name: Schmidt subspace theorem
msc: [11J87, 11J68]
uses: [07YQ1C78G0, 0CRMGPAH83, 0MNF0BYFQC]
about: [0MNF0BYFQC, 07YQ1C78G0]
sources:
  - url: https://en.wikipedia.org/wiki/Subspace_theorem
same_as:
  wikidata: Q7632041
  wikipedia: Subspace theorem
---

## Statement

Let $n \ge 2$, and let $L_1, \dots, L_n$ be [[linearly independent|0CRMGPAH83]] [[linear forms|07YQ1C78G0]] in $n$ variables whose coefficients are real [[algebraic numbers|0MNF0BYFQC]].
Let $\varepsilon > 0$.
For $x = (x_1, \dots, x_n) \in \Z^n$ write $\norm{x} = \max_i \abs{x_i}$.
Then the set of nonzero $x \in \Z^n$ with
$$\abs{L_1(x) \, L_2(x) \cdots L_n(x)} < \norm{x}^{-\varepsilon}$$
is contained in the union of finitely many proper linear subspaces of $\Q^n$.

## 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|R9JZA3G2NX]].
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.

## Explanations

### L1 human

Roth's theorem says that an algebraic number has few very good rational approximations.
The subspace theorem says the same for several algebraic linear forms at once.
The integer points where the product of the forms is very small do not spread out: they all lie on finitely many planes.

### L2 human

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.