Ostrowski Numeration, Alternating Expansions\\ and Inhomogeneous Linear Approximation

Given an irrational slope $α$ and a real intercept $β$, the inhomogeneous linear approximation problem asks for integers $m$ for which $\norm{mα-β}$ is small compared with $\abs{m}$. Building on Ostrowski's numeration system, we give a self-contained treatment of its alternating (dual) counterpart,