Bounding the Largest Inhomogeneous Approximation Constant

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, $$\rho(\alpha) \geq \cfrac{1}{4\sqrt{3}+2}=\cfrac{1}{8.9282\ldots}.$$