An upper bound on the inhomogeneous approximation constants

For an irrational real $\alpha$ and $\gamma \in ̸\mathbb{Z}+\mathbb{Z}\alpha$ it is well known that $$\underset{\mathrm{\mid }n\mathrm{\mid }\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\mathrm{\mid }n\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }n\alpha -\gamma \mathrm{\mid }\mathrm{\mid }\le \frac{1}{4}\mathrm{.}$$ If the partial quotients, $a_i,$ in the negative `round-up’ continued fraction expansion of $\alpha$ have $R:={\mathrm{lim\hspace{0.17em}inf}}_{i\to \mathrm{\infty }}{a}_i$ odd, then the 1/4 can be replaced by $$\frac{1}{4}(1-\frac{1}{R})(1-\frac{1}{R^2}),$$ which is optimal. The optimal bound for even $R\ge 4$ was already known.