On Convergents of Proper Continued Fractions

2024/12/06 by Langeveld, Niels, Ralston, David
#11J70 #37A44 #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2412.05077

Abstract

Proper continued fractions are generalized continued fractions with positive integer numerators a_i and integer denominators with b_i≥ a_i. In this paper we study the strength of approximation of irrational numbers to their convergents and classify which pairs of integers p,q yield a convergent p/q to some irrational x. Notably, we reduce the problem to finding convergence only of index one and two. We completely classify the possible choices for convergents of odd index and provide a near-complete classification for even index. We furthermore propose a natural two-dimensional generalization of the classical Gauss map as a method for dynamically generating all possible expansions and establish ergodicity of this map.

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