Bounding the Largest Inhomogeneous Approximation Constant

TL;DR

This paper establishes optimal lower bounds for the inhomogeneous approximation constant of irrational numbers based on the liminf of their partial quotients, providing precise bounds for various cases.

Contribution

It derives the best possible lower bounds for the inhomogeneous approximation constant in terms of the partial quotients' liminf, extending understanding of inhomogeneous Diophantine approximation.

Findings

For R ≥ 3, ρ(α) ≥ 1/(6√3+8) ≈ 1/18.39.

For R ≥ 4, ρ(α) ≥ 1/(4√3+2) ≈ 1/8.93.

Bounds are optimal when R is even and asymptotically precise when R is odd.

Abstract

For a given irrational number $\alpha$ and a real number $\gamma$ in $(0,1)$ one defines the two-sided inhomogeneous approximation constant \begin{equation*} M(\alpha,\gamma):=\liminf_{|n|\rightarrow\infty}|n| ||n\alpha-\gamma||, \end{equation*} and the case of worst inhomogeneous approximation for $\alpha$ \begin{equation*} \rho(\alpha):=\sup_{\gamma\notin\mathbb{Z}+\alpha\mathbb{Z}}M(\alpha,\gamma). \end{equation*} We are interested in lower bounds on $\rho(\alpha)$ in terms of $R:=\liminf_{i\rightarrow\infty}a_i,$ where the $a_i$ are the partial quotients in the negative (i.e.\ the `round-up') continued fraction expansion of $\alpha$. We obtain bounds for any $R\geq 3$ which are best possible when $R$ is even (and asymptotically precise when $R$ is odd). In particular when $R\geq 3$ $$ \rho(\alpha)\geq \cfrac{1}{6\sqrt{3}+8}=\cfrac{1}{18.3923\dots}, $$ and when $R\geq 4$, optimally, $$ \rho(\alpha) \geq \cfrac{1}{4\sqrt{3}+2}=\cfrac{1}{8.9282\ldots}. $$