Schmidt subspace theorem

Let $L_1,\dots ,L_n$ be linearly independent linear forms in $n$ variables with algebraic coefficients. For every $\epsilon >0$, the nonzero $\mathbf{x}\in {\mathbb{Z}}^n$ satisfyinglie in finitely many proper linear subspaces of ${\mathbb{Q}}^n$. There are variants involving a finite set of places of a number field.

If $\alpha$ is a real algebraic irrational number and $\epsilon >0$, thenhas only finitely many reduced rational solutions $p/q$.

Let $\alpha$ be an algebraic irrational number, let $S$ be a finite set of primes, and let $\epsilon >0$. There are only finitely many reduced fractions $p/q$ whose denominator is S-smooth number and which satisfyThis is the denominator-restricted consequence of the finite-place Schmidt subspace theorem.