Distance between cubics and rationals

Abstract

We investigate the following problem: what is the smallest possible distance between a cubic irrational $\xi$ and a rational number $p/q$ in terms of the height $H(\xi)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of all pairs $(u,v)$ of positive real numbers such that $|\xi - p/q| > cH^{-u}(\xi)q^{-v}$ for all cubic irrationals $\xi$ and rationals $p/q$. First, we transform this problem into one about the root separation of cubic polynomials. Second, under the assumption of the famous abc-conjecture, we give an almost complete description of $D_{3,1}$. Namely, the points $(u,v)$ with $2\le v\le 3$ that lie in the interior of $D_{3,1}$ are characterised by the inequality $u> 10-3v$. Assuming only the weaker Hall conjecture, we also obtain nontrivial results about the shape of $D_{3,1}$, although these are not as strong as those derived from the abc-conjecture. Finally, we discuss an analogue of the set $D_{3,1}$ in function fields where we are able to give an almost complete description unconditionally.