Convergents and best approximates

Experts agree that if a generalization for finding the convergents of the continued fraction of a real number existed, the Littlewood conjecture would not remain unsolved. We will consider two natural generalizations of the convergents of the continued fraction of a single real number, based on Lagrange's characterization in terms of best approximates of the second kind. We consider a generalization of a convergent using pivots, and a generalization called an TV-best approximate. We will clarify the relation between these two generalizations by showing that every TVbest approximate is a convergent. Moreover, we show how the notion of an TV-best approximate is sensitive to choice of norm TV by exhibiting an example of an TV-best approximate with respect to one norm which is not an TV-best approximate with respect to a different norm. Finally, we show TV-best approximates have the nearest integer property that convergents have by definition.