Abstract: For an irrational real $\alpha$ and $\gamma\not \in \mathbb Z + \mathbb Z\alpha$ it is well known that
$M(\alpha,\gamma):= \liminf_{|n|\rightarrow \infty} |n| ||n\alpha -\gamma || \leq \frac{1}{4}.$
In this talk, I will discuss the upper and lower bounds which will depend on whether $R:=\liminf_{i\rightarrow\infty}a_i$ is odd or even, where $a_i$ are the partial quotients in the negative (i.e. the `round-up') continued fraction expansion of $\alpha$. We will see upper bounds of the form $M(\alpha,\gamma)\leq C_0(R)$ with $C_0(R)$ optimal for $R\geq 3$. Then the lower bounds, that there is always a $\gamma$ such that $M(\alpha,\gamma)\geq C_1(R)$ with $C_1(R)$ optimal when $R$ is even, and asymptotically optimal when $R$ is odd. In particular, for any $R\geq3$ there is a $\gamma\notin \mathbb Z + \mathbb Z\alpha$ with
$$M(\alpha,\gamma)\geq\frac{1}{6\sqrt{3}+8}=\frac{1}{18.3923...}.$$
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