Distance Between Cubics and Rationals

Abstract We investigate the following problem: what is the smallest possible distance between a cubic irrational ξ and a rational number p / q in terms of the height H () H (ξ) and q? More precisely, we consider the set D₃, ₁ D 3, 1 consisting of all pairs (u, v) of positive real numbers such that | - p/q| > cH^-u () q^-v | ξ - p / q | > c H - u (ξ) q - v for all cubic irrationals ξ 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 D₃, ₁ D 3, 1. Namely, the points (u, v) with 2 v 3 2 ⩽ v ⩽ 3 that lie in the interior of D₃, ₁ D 3, 1 are characterised by the inequality u> 10-3v u > 10 - 3 v. Assuming only the weaker Hall conjecture, we also obtain nontrivial results about the shape of D₃, ₁ D 3, 1, although these are not as strong as those derived from the abc-conjecture. Finally, we discuss an analogue of the set D₃, ₁ D 3, 1 in function fields where we are able to give an almost complete description unconditionally.