The Subspace Theorem

We start with some history. Let $\alpha$ be a real irrational algebraic number of degree $d$ and let $\kappa >0$. In 1909, Thue [28] proved that for any $\kappa >\frac{1}{2}d+1$, the inequality

$\vert\alpha -\frac{x}{y}\vert< \max (\vert x\vert,\vert y\vert)^{-\kappa}$ (6)

has only finitely many solutions in pairs of integers $(x,y)$ with $y>0$. After improvements of Thue's result by Siegel, Gel'fond and Dyson, in 1955 Roth [22] proved that (6) has only finitely many solutions in pairs of integers $(x,y)$ with $y>0$ already when $\kappa >2$. This lower bound $2$ is best possible, since by a result of Dirichlet from 1842, for any irrational real number $\alpha$ there are infinitely many pairs of integers $(x,y)$ with

$\vert\alpha -\frac{x}{y}\vert< y^{-2},\quad y>0\, .$

In a sequence of papers from 1965-1972, W.M. Schmidt proved a far reaching higher dimensional generalization of Roth's theorem, now known as the Subspace Theorem. For a full proof of the Subspace Theorem as well as of the other results mentioned above we refer to Schmidt's lecture notes [25]. Below we have stated the version of the Subspace Theorem which is most convenient for us. We define the norm of ${\bf x}=(x_1,\ldots,x_n)\in{\mathbb{Z}}^n$ by $\Vert{\bf x}\Vert :=\max (\vert x_1\vert,\ldots,\vert x_n\vert)$.

Subspace Theorem (Schmidt). Let

$$L_1=\alpha_{11}X_1+\cdots+\alpha_{1n}X_n,\ldots, L_n=\alpha_{n1}X_1+\cdots+\alpha_{nn}X_n$$

be $n$ linearly independent linear forms with real or complex algebraic coefficients $\alpha_{ij}$. Let $c_1,\ldots,c_n$ be reals with

$$c_1+\cdots+c_n <0\, .$$

Consider the system of inequalities

$\vert L_1({\bf x})\vert<\Vert{\bf x}\Vert^{c_1},\ldots, \vert L_n({\bf x})\vert<\Vert{\bf x}\Vert^{c_n}$ (7)

to be solved simultaneously in integer vectors ${\bf x}\in{\mathbb{Z}}^n$.
Then there are proper linear subspaces $T_1,\ldots,T_t$ of ${\mathbb{Q}}^n$ such that the set of solutions of (7) is contained in $T_1\cup\cdots\cup T_t$.

Roth's Theorem follows by taking $n=2$, $L_1=X_1-\alpha X_2$, $L_2=X_2$, $c_1= 1-\kappa$, $c_2=1$. Thus, if ${\bf x}=(x,y)$ is a solution of (6) with $y\not =0$ then ${\bf x}$ also satisfies (7).

We give another example to illustrate the Subspace Theorem. Consider the system

$$\left\{ \begin{array}{rcl} \vert x_1+\sqrt{2}x_2+\sqrt{3}x_3... ..._1\vert,\vert x_2\vert,\vert x_3\vert)^{-1} \end{array}\right.$$ (8)

The Pell equation $x_1^2-2x_2^2=1$ has infinitely many solutions in positive integers $x_1,x_2$. It is easy to see that if $(x_1,x_2)$ is a solution of the Pell equation with $x_2\geqslant 2$ and if $x_3=0$, then $(x_1,x_2,x_3)$ is a solution of (8). Thus, the subspace $x_3=0$ contains infinitely many solutions of (8). One can prove something more precise than predicted by the Subspace Theorem, that is, that (8) has only finitely many solutions with $x_3\not= 0$.

In 1977, Schlickewei [23] proved a so-called p-adic version of the Subspace Theorem, involving, apart from the usual absolute value, a finite number of p-adic absolute values. Given a rational number $\alpha\in{\mathbb{Q}}$ and a prime number $p$, we define $\vert\alpha \vert _p:= p^{-w}$ where $w$ is the exponent such that $\alpha =p^w\cdot a/b$ with $a,b$ integers not divisible by $p$. For instance, $\vert 9/8\vert _2=8$ and $\vert 9/8\vert _3=1/9$. The $p$-adic absolute value $\vert\cdot \vert _p$ defines a metric on ${\mathbb{Q}}$. By taking the metric completion we obtain a field ${\mathbb{Q}}_p$. Let ${\mathbb{C}}_p$ denote the algebraic closure of ${\mathbb{Q}}_p$. The $p$-adic absolute value can be extended uniquely to ${\mathbb{C}}_p$. To get a uniform notation, we write $\vert\cdot \vert _{\infty}$ for the usual absolute value $\vert\cdot \vert$, and ${\mathbb{C}}_{\infty}$ for ${\mathbb{C}}$. We call $\infty$ the infinite prime of ${\mathbb{Q}}$. We will use the index $p$ to indicate either $\infty$ or a prime number. Then we get:
p-adic Subspace Theorem (Schlickewei). Let $S=\{\infty ,p_1,\ldots,p_t\}$ consist of the infinite prime and a finite number of primes numbers. For $p\in S$, let

$$L_{1p}=\alpha_{11p}X_1+\cdots+\alpha_{1np}X_n,\ldots, L_{np}=\alpha_{n1p}X_1+\cdots+\alpha_{nnp}X_n$$

be linearly independent linear forms with coefficients $\alpha_{ijp}\in{\mathbb{C}}_p$ which are algebraic over ${\mathbb{Q}}$. Further, let $c_{ip}$ ( $i=1,\ldots,n$, $p\in S$) be reals satisfying

$$\sum_{p\in S}\sum_{i=1}^n c_{ip} < 0\, .$$

Consider the system of inequalities
$\vert L_{ip}({\bf x})\vert _p\,\leqslant \,\Vert{\bf x}\Vert^{c_{ip}}\quad (p\in S,\,\, i=1,\ldots,n)$ (9)

to be solved simultaneously in ${\bf x}\in{\mathbb{Z}}^n$.
Then there are proper linear subspaces $T_1,\ldots,T_t$ of ${\mathbb{Q}}^n$ such that the set of solutions of (9) is contained in $T_1\cup\cdots\cup T_t$.
There is a further generalization of this result, which we shall not state, dealing with systems of inequalities to be solved in vectors consisting of integers from a given algebraic number field. This generalization has a wide range of applications, such as finiteness results for Diophantine equations of the type considered in the previous sections, finiteness results for all sorts of Diophantine inequalities, transcendence results, finiteness results for integral points on surfaces, etc.

As an illustration, we consider the equation

$2^{z_1}+2^{z_2}-11^{z_3}=1$ (10)

to be solved in $z_1,z_2,z_3\in{\mathbb{Z}}$. It is easy to see that (10) has only solutions with non-negative $z_1,z_2,z_3$. Notice that $(2^{z_1},2^{z_2},11^{z_3})$ is a solution of $x_1+x_2-x_3=1$ in $x_1,x_2,x_3\in\Gamma =\{ 2^u11^v:\, u,v\in{\mathbb{Z}}\}$. Hence equation (10) may be viewed as a special case of (2).
Put $x_1=2^{z_1}$, $x_2=2^{z_2}$, $x_3=11^{z_3}$, $\xi =\log x_1/\log x_3$, $\eta =\log x_2/\log x_3$, ${\bf x}=(x_1,x_2,x_3)$. Then $\Vert {\bf x}\Vert=x_3$ and $0\leqslant \xi ,\eta\leqslant 1$. Hence there are $k,l\in\{ 0,1,2\}$ such that $\frac{k}{3}\leqslant \xi\leqslant \frac{k+1}{3}$ and $\frac{l}{3}\leqslant \eta\leqslant \frac{l+1}{3}$. We consider those solutions with fixed values of $k,l$. Notice that these solutions satisfy the inequalities

$$\vert x_1+x_2-x_3\vert _{\infty}\leqslant \Vert{\bf x}\Vert^0,\qu... ...t^{(k+1)/3},\quad \vert x_2\vert _{\infty}\leqslant \Vert{\bf x}\Vert^{(l+1)/3}$$

$$\vert x_1\vert _2\leqslant \Vert{\bf x}\Vert^{-k/3},\quad \vert x... ...t \Vert{\bf x}\Vert^{-l/3},\quad \vert x_3\vert _2\leqslant \Vert{\bf x}\Vert^0$$

$$\vert x_1\vert _{11}\leqslant \Vert{\bf x}\Vert^0,\quad \vert x_2... ...rt{\bf x}\Vert^0,\quad \vert x_3\vert _{11}\leqslant \Vert{\bf x}\Vert^{-1}\, .$$

This system is a special case of (9), and since the sum of the exponents is $-1/3<0$ we can apply the p-adic Subspace Theorem with $n=3$.
Taking into consideration the possibilities for $k,l$, we see that ${\bf x}=(x_1,x_2,x_3)$ $= (2^{z_1},2^{z_2},11^{z_3})$ is contained in the union of finitely many proper linear subspaces of ${\mathbb{Q}}^3$. Considering the solutions in a single subspace, we can eliminate one of the variables $x_1,x_2,x_3$ and obtain an equation of the same type as (10), but in only two variables. Applying again the p-adic Subspace Theorem but now with $n=2$, we obtain that the solutions lie in finitely many one-dimensional subspaces, etc. Eventually we obtain that (10) has only finitely many solutions.

In 1989, Schmidt [26] obtained a quantitative version of his Subspace Theorem, giving an explicit upper bound for the number of subspaces $t$. Since then, his result has been refined and improved in several directions. In particular Schlickewei obtained quantitative versions of his p-adic Subspace Theorem which enabled him to prove weaker versions of Theorem 1 with an upper bound depending on $r,n$ and other parameters and of Schmidt's theorem on linear recurrences with an upper bound depending on $k$ and other parameters. Finally, Schlickewei and the author [7] managed to prove a quantitative version of the p-adic Subspace Theorem with unknowns taken from the ring of integers of a number field which was strong enough to imply the upper bounds mentioned in the previous sections. We will not give the rather complicated statement of this result.

By using a suitable specialization argument from algebraic geometry one may reduce Theorem 1 to the case that $a_1,\ldots,a_n$ and the group $\Gamma$ are contained in an algebraic number field, and then subsequently one may reduce equation (2) to a finite number of systems (9) by a similar argument as above. By applying the quantitative p-adic Subspace Theorem to each of these systems and adding together the upper bounds for the number of subspaces for each system, one obtains an explicit upper bound for the number of subspaces containing the solutions of (2). Considering the solutions of (2) in one of these subspaces, then by eliminating one of the variables one obtains an equation of the shape (2) in $n-1$ variables to which a similar argument can be applied. By repeating this, Theorem 1 follows.
The proof of Schmidt's theorem on linear recurrence sequences has a similar structure, but there the argument is much more involved.