Two-Dimensional Analogs of the Minkowski ?(x) Function

A Two-Dimensional Minkowski ?(x) Function

A one-to-one continuous function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is shown there exists a natural class of pairs of cubic irrational numbers in the same cubic number field that are mapped to pairs of rational numbers, in analog to ?(x) mapping quadratic irrationals on the unit interval to rational numbers on the unit interval. It is also shown that this new function satisfies an analog to the fact that ?(x), while increasing …

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

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 t…

关于到尖点距离的Minkowski型定理：类数为1的情形

A Minkowski-type theorem on distances to cusps: the class number one case

本文通过建立二维刚性adèle空间与H^n中点的类比，将关于Roy–Thunder极小值的Minkowski型定理转化为H^n中到尖点距离的定理，在类数为1的情形下给出新结果。

On Minkowski type question mark functions associated with even or odd continued fractions

We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are Hölder continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.

Multidimensional continued fractions and a Minkowski function

The Minkowski Question Mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Monkemeyer continued fraction algorithm with an appropriate tent map.