# Generalizing the Minkowski Question Mark Function to a Family of Multidimensional Continued Fractions

## TL;DR

This paper extends the Minkowski question mark function to a family of multidimensional continued fractions called TRIP maps, demonstrating that most of these analogs are singular functions, thus generalizing a key number theoretic concept.

## Contribution

It introduces a natural generalization of the Minkowski question mark function to 216 TRIP maps and proves their singularity in most cases, expanding understanding of multidimensional continued fractions.

## Key findings

- 96 TRIP map analogs are singular functions.
- An additional 60 are singular under ergodicity assumptions.
- The generalization encompasses almost all known multidimensional continued fractions.

## Abstract

The Minkowski question mark function, maping the unit interval to itself, is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus the question mark function is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.