Rational Approximations to Certain Algebraic Numbers

TL;DR

This paper proves that for certain algebraic numbers, there exists an effective constant ensuring a lower bound on approximation quality, implying their continued fraction partial quotients are bounded, supporting conjectures about their approximation properties.

Contribution

The paper establishes effective bounds for approximation of specific algebraic numbers, demonstrating their partial quotients are bounded, which advances understanding of their Diophantine approximation properties.

Findings

Existence of an effective constant C for algebraic numbers ensuring |p - qθ| > Cq^{-1}.

Boundedness of the sequence of partial quotients for these algebraic numbers.

Supports conjecture that certain algebraic numbers have unbounded partial quotients, but with proven bounds.

Abstract

W.M.Schmit[11] conjectured that for any$\;\theta$ with deg$\;\theta\geq 3,$ there is no constant$\;C=C(\theta)$ so that$\;|p-q\theta|>Cq^{-1}$ for every rationa$\;p/q.$ [12,p26] states that the computations of the first several thousand partial quotients for such numbers as$\;\sqrt[3]{2}$ and$\;\sqrt[3]{3}$ support the conjecture that the sequence of partial quotients is unbounded. In this paper, applying Dirichlet's approximation theorem to certain algebraic numbers$\;\theta,$ e.g.$\;\theta=\sqrt[n]{d},d\in N,n\geq 3,d>0;$ $\;\theta^{3}+b_{1}\theta-b_{0}=0,b_{0}>0;$ $\;\theta^{4}+b_{2}\theta^{2}-b_{0}=0,\;b_{0}>0.$ We proved that there exists a effective constant$\;C=C(\theta)$ such that$\;|p-q\theta|>Cq^{-1}$ for all$\;p/q.$ Our theorem shows their sequence of partial quotients can not be unbounded.