For a given irrational number \( \alpha \) and a real number \( \gamma \in (0,1) \) one defines the two-sided inhomogeneous approximation constant

\[
M(\alpha,\gamma):=\liminf_{|n|\to\infty}\|n\alpha-\gamma\|,
\]

and the case of worst inhomogeneous approximation for \( \alpha \)

\[
\rho(\alpha):=
\sup_{\gamma \notin \alpha\mathbb{Z}+\mathbb{Z}}
M(\alpha,\gamma).
\]

We are interested in lower bounds on \( \rho(\alpha) \) in terms of
\( R:=\liminf_{i\to\infty} a_i \), 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 \( R \ge 3 \)

\[
\rho(\alpha)\ge
\frac{1}{6\sqrt{3}+8}
=
\frac{1}{18.3923\ldots},
\]

and when \( R \ge 4 \), optimally,

\[
\rho(\alpha)\ge
\frac{1}{4\sqrt{3}+2}
=
\frac{1}{8.9282\ldots}.
\]