Ostrowski Numeration, Alternating Expansions\\ and Inhomogeneous Linear Approximation

Abstract

Given an irrational slope $\alpha$ and a real intercept $\beta$, 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, whose weights are the signed convergent denominators $(-1{)}^k{q}_k$, and show that with $n$ digits it represents exactly the integers of an explicit interval of length $q_n+q_{n-1}$, compared with $q_n$ for Ostrowski's system. We pair the absolute and the alternating integer systems with, respectively, Ostrowski's classical alternating expansion and an absolute expansion of real numbers in the linear approximation coefficients ${\theta }_k=q_k\alpha -p_k$. These expansions yield explicit algorithms that produce, for the one--sided and the two--sided (forward and backward) variants of the problem, sequences of approximates $A_n$ with $\abs{A_n}\le q_n$ and error at most $\abs{θ_{n-1}}$. For the one--sided variants we prove optimality: the $n$--th term is the best one--sided approximate within the $n$--th range of the alternating system.