Solving for best inhomogeneous linear approximates

Our goal is to finally settle a persistent problem in Diophantine Approximation, that of finding best inhomogeneous linear approximates. Classical results from the theory of continued fractions solve the special homogeneous case in the form of a complete sequence of normal approximates. Real expansions that allow the notion of normality to percolate into the inhomogeneous setting will provide us with the general solution.