Solving for best linear approximates

TL;DR

This paper advances the theory of Diophantine Approximation by developing new numeration systems and expansions to find best linear approximates in both homogeneous and inhomogeneous cases.

Contribution

It introduces novel numeration systems and real expansions that extend the concept of normality to the inhomogeneous setting, addressing a longstanding problem.

Findings

Established a method for inhomogeneous best linear approximation

Extended continued fraction techniques to more general settings

Provided a framework for future research in Diophantine Approximation

Abstract

Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the form of a sequence of normal approximates. We develop numeration systems and real expansions allowing this notion of normality to percolate into the general inhomogeneous setting.