Distance between cubics and rationals
Dmitry Badziahin
TL;DR
The paper studies the minimal distance between cubic irrationals $\xi$ and rationals $p/q$ in terms of the heights, formalizing the problem via the set $D_{3,1}$. It reduces the question to root-separation bounds for cubic polynomials: if $\mathrm{Sep}(Q) \ge c B^{-2-s} A^{-2-t}$ for all irreducible cubics $Q$, then $\left(\frac{2+s}{s+t}, 2+\frac{s}{s+t}\right) \in D_{3,1}$, linking Diophantine approximation to polynomial geometry. Conditional results under the Hall conjecture yield explicit parametric families of $(u,v)$ within $D_{3,1}$ approaching $\left(\tfrac{5}{2},\tfrac{5}{2}\right)$, while the stronger $abc$-conjecture implies a near-complete description of the interior of $D_{3,1}$, notably the region $2<v\le 3$ with $u>10-3v$. The paper also derives Thue-equation corollaries, constructs an explicit infinite family of close cubic irrationals, and presents an unconditional function-field analogue, highlighting how the real and function-field settings interact with deep conjectures. Overall, this work ties Diophantine approximation to root-separation phenomena and conjectural bounds, yielding both conditional structure theorems and concrete examples.
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 $H(ξ)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of all pairs $(u,v)$ of positive real numbers such that $|ξ- p/q| > cH^{-u}(ξ)q^{-v}$ 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 $D_{3,1}$. 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.