Tong's spectrum for Rosen continued fractions

TL;DR

This paper establishes optimal bounds for approximation coefficients in Rosen continued fractions and explores their statistical properties regarding poor approximations.

Contribution

It provides the best possible bounds for approximation coefficients in Rosen continued fractions and analyzes their metrical behavior for large blocks of bad approximations.

Findings

Derived optimal upper bounds for approximation coefficients.

Obtained metrical results for large blocks of poor approximations.

Enhanced understanding of Diophantine approximation in Rosen fractions.

Abstract

The Rosen fractions are an infinite set of continued fraction algorithms, 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 approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.