On the arithmetic of multidimensional continued fractions

TL;DR

This paper extends Gosper's algorithm to develop an arithmetic framework for multidimensional continued fractions, enabling operations like addition and multiplication, with experimental analysis of the algorithms' behavior.

Contribution

The paper introduces a novel arithmetic for multidimensional continued fractions by extending existing algorithms and defining transformations such as M"obius and bilinear transforms.

Findings

Algorithms for MCF arithmetic are effective in performing sums and products.

Experimental results show the behavior and stability of the proposed algorithms.

The approach generalizes previous methods to higher dimensions.

Abstract

The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the M\"obius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms.