Combinatorial properties of multidimensional continued fractions

TL;DR

This paper explores the combinatorial aspects of multidimensional continued fractions, providing a new interpretation of their convergents through tiling counts, extending classical results to higher dimensions.

Contribution

It introduces a novel combinatorial interpretation of multidimensional continued fractions' convergents, generalizing classical continued fraction properties.

Findings

Convergent properties linked to specific tilings

Extension of classical continued fraction results to multiple dimensions

New combinatorial framework for multidimensional continued fractions

Abstract

The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.