An upper bound on the inhomogeneous approximation constants

Abstract

For an irrational real $\alpha$ and $\gamma \notin \mathbb{Z}+\mathbb{Z}\alpha$ it is well known that $$\underset{|n|\to \infty }{\mathrm{liminf}}|n|||n\alpha -\gamma ||\le \frac{1}{4}.$$ If the partial quotients, $a_i,$ in the negative `round-up' continued fraction expansion of $\alpha$ have $R:={\mathrm{liminf}}_{i\to \infty }{a}_i$ 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 $R\ge 4$ was already known.