# Dirichlet's theorem with an arbitrarily small constant for algebraic numbers of degree $d \geq 3$

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

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

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.