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Dirichlet's theorem on diophantine approximation asserts that, for every irrational real number $\alpha$, there are infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1, q > 0$ such that

$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^2}.$$

It can be shown that the numerator $1$ can be replaced with any constant $> 1/\sqrt{5}$ (Hurwitz's theorem).

My question is, if we restrict to real algebraic numbers $\alpha$ with degree $d \geq 3$, can we replace the $1$ in the numerator of Dirichlet's theorem with an arbitrarily large small number? Indeed, we ask:

Let $\alpha$ be an algebraic number of degree $d \geq 3$ and let $c > 0$. Do there exist infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1$ and $q > 0$ such that

$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{c}{q^2}?$$

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We expect this to be true for all $\alpha$ (an algebraic number of degree at least three), but we don't even know if such an $\alpha$ exists. See Page 366 in the book Hindry-Silverman: Diophantine Geometry - An Introduction.

I took the above information from Is any particular algebraic number known to have unbounded continued fraction coefficients? One can also learn from here that the question was originally asked by Khintchine (1935).

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