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

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

2004-05-24

arxiv.org/abs/math/0405446v1

Mathematics

Number Theory

50 pages, 13 figures, for associated applet, see http://wso.williams.edu/~amarder/applets/

Scientific paper

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.

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