Bounding the Largest Inhomogeneous Approximation Constant

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(α,γ):=\liminf_{|n|\rightarrow\infty}|n| ||nα-γ||, \end{equation*} and the case of worst inhomogeneous approximation for $\alpha$ \begin{equation*} ρ(α):=\sup_{γ\notin\mathbb{Z}+α\mathbb{Z}}M(α,γ). \end{equation*} We are interested in lower bounds on $\rho (\alpha )$ in terms of $R:={\mathrm{liminf}}_{i\to \infty }{a}_i,$ 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 $R\ge 3$ which are best possible when $R$ is even (and asymptotically precise when $R$ is odd). In particular when $R\ge 3$ $$\rho (\alpha )\ge \frac{1}{6\sqrt{3}+8}=\frac{1}{18.3923\dots },$$ and when $R\ge 4$, optimally, $$\rho (\alpha )\ge \frac{1}{4\sqrt{3}+2}=\frac{1}{8.9282\dots }.$$