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 \(2\leqslant v\leqslant 3\) 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.
