Tong’s spectrum for Rosen continued fractions

Abstract

In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of k consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algo- rithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approxima- tion coecients. We also obtain metrical results for large blocks of bad approximations.