Distance Between Cubics and Rationals

We investigate the following problem: what is the smallest possible distance between a cubic irrational ξ ξ and a rational number p / q in terms of the height H(ξ ) 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 |ξ - p/q| > cH⁻ᵘ(ξ )q⁻ᵛ | ξ - p / q | > c H - u ( ξ ) q - v for all cubic irrationals ξ 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.