= Solution
The <Roth theorem> states that if $\alpha$ is a real algebraic irrational number, then for every $\varepsilon>0$ there are only finitely many reduced fractions $p/q$ satisfying
$$
\left|\alpha-\frac pq\right|<q^{-2-\varepsilon}.
$$
To derive this from the <Schmidt subspace theorem>, take
$$
L_1(X,Y)=X-\alpha Y,
\qquad
L_2(X,Y)=Y.
$$
These forms are linearly independent. For a solution $(p,q)$ with $q$ large, $|p|\asymp q$, so $H(p,q)\asymp q$, while
$$
|L_1(p,q)L_2(p,q)|
=|p-\alpha q|q
=q^2\left|\alpha-\frac pq\right|
<q^{-\varepsilon}.
$$
After slightly decreasing $\varepsilon$, 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.
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