An upper bound on the inhomogeneous approximation constants

TL;DR

This paper establishes an optimal upper bound on inhomogeneous approximation constants for irrational numbers, refining previous bounds based on the parity of the lim inf of partial quotients in their continued fraction expansion.

Contribution

It provides a new optimal upper bound for the inhomogeneous approximation constant when the lim inf of partial quotients is odd, extending known results for even cases.

Findings

For odd R, the bound is rac{1}{4}(1-rac{1}{R})(1-rac{1}{R^2})

The bound is optimal for odd R

Previous bounds for even R are confirmed to be optimal

Abstract

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.