  For an irrational real $\alpha$ and $\gamma\not \in \mathbb Z + \mathbb
Z\alpha$ it is well known that $$ \liminf_{|n|\rightarrow \infty} |n| ||n\alpha
-\gamma || \leq \frac{1}{4}. $$ If the partial quotients, $a_i,$ in the
negative `round-up' continued fraction expansion of $\alpha$ have
$R:=\liminf_{i\rightarrow \infty}a_i$ odd, then the 1/4 can be replaced by $$
\frac{1}{4}\left(1-\frac{1}{R}\right)\left(1-\frac{1}{R^2}\right), $$ which is
optimal. The optimal bound for even $R\geq 4$ was already known.
