Let $ \alpha$ be a real algebraic number. It is easy to see that if $ \deg(\alpha) = 2$ , that for there exists a number $ c(D(\alpha))$ , where $ D(\alpha)$ is the discriminant of the primitive integral quadratic polynomial whose root is $ \alpha$ , such that

$ $ \displaystyle \left \lvert \alpha – \frac{p}{q} \right \rvert > \frac{c(D(\alpha))}{q^2}$ $

for all rational numbers $ p/q$ with $ q > 0$ and $ \gcd(p,q) = 1$ . In view of Dirichlet’s theorem, this is the best possible result.

In 1958 Klaus Roth proved the following stunning theorem, as a culmination of the approach invented by Axel Thue in 1909 and developed by many subsequent authors including Siegel and Dyson. He proved that for any algebraic number $ \alpha$ and $ \varepsilon > 0$ there exists a number $ c(\alpha, \varepsilon)$ such that for rational numbers $ p/q$ with $ q > 0$ and $ \gcd(p,q) = 1$ we have

$ $ \displaystyle \left \lvert \alpha – \frac{p}{q} \right \rvert > \frac{c(\alpha, \varepsilon)}{q^{2 + \epsilon}}.$ $

The problem is that the dependence of $ c(\alpha, \varepsilon)$ on either parameter is ineffective. Thus in many applications one has to use a qualitatively worse but effective version of Roth’s theorem to get results. Lang also conjectured that one can replace the $ q^\varepsilon$ term in the denominator with $ (\log q)^{1 + \varepsilon}$ . Indeed, if we take $ \varepsilon = 1$ we can ‘remove’ the dependence on $ \varepsilon$ completely.

I am asking about what is expected conjecturally, in view of the quality of result in the quadratic case. In particular, does one expect to be able to find a number $ c(\alpha)$ such that

$ $ \displaystyle \left \lvert \alpha – \frac{p}{q} \right \rvert > \frac{c(\alpha)}{(q \log q)^2}$ $

for all $ p/q \in \mathbb{Q}, \gcd(p,q) = 1, q > 0$ , and such that $ c(\alpha)$ can be made to depend at most polynomially on the height of the algebraic number $ \alpha$ ?