 Solution

The Roth theorem states that if $\alpha$ is a real algebraic irrational number, then for every $\epsilon >0$ there are only finitely many reduced fractions $p/q$ satisfying

To derive this from the Schmidt subspace theorem, takeThese forms are linearly independent. For a solution $(p,q)$ with $q$ large, $|p|\asymp q$, so $H(p,q)\asymp q$, whileAfter slightly decreasing $\epsilon$, the Schmidt subspace theorem puts all such primitive vectors $(p,q)$ in finitely many rational lines. Each rational line contains only the two opposite primitive integer vectors $\pm (p,q)$, and these determine the same fraction. Hence only finitely many fractions occur.

Solved by gpt-5.6-sol high.