Two-Dimensional Analogs of the Minkowski ?(x) Function
Number Theory
2007-05-23 v1 Classical Analysis and ODEs
Abstract
Two generalizations of the Minkowski ?(x) function are given. As ?(x) maps quadratic irrationals to rational numbers, it is shown that both generalizations send natural classes of pairs of cubic irrational numbers in the same cubic number field to pairs of rational numbers. It is also shown that these functions satisfy an analog to the fact that ?(x), while continuous and increasing, has derivative zero almost everywhere. Both extend earlier work of Beaver-Garrity on the Farey-Bary map.
Cite
@article{arxiv.math/0405446,
title = {Two-Dimensional Analogs of the Minkowski ?(x) Function},
author = {Andrew Marder},
journal= {arXiv preprint arXiv:math/0405446},
year = {2007}
}
Comments
50 pages, 13 figures, for associated applet, see http://wso.williams.edu/~amarder/applets/