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arXiv · 2412.05077

On Convergents of Proper Continued Fractions

Niels Langeveld·David Ralston
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Abstract

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. 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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Niels Langeveld, David Ralston. 2024-12-06. On Convergents of Proper Continued Fractions. https://arxiv.org/abs/2412.05077

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