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