TY - JOUR

T1 - Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms

AU - Dasaratha, Krishna

AU - Flapan, Laure

AU - Garrity, Thomas

AU - Lee, Chansoo

AU - Mihaila, Cornelia

AU - Neumann-Chun, Nicholas

AU - Peluse, Sarah

AU - Stoffregen, Matthew

N1 - Funding Information:
The authors thank the National Science Foundation for their support of this research via grant DMS-0850577.

PY - 2014/8

Y1 - 2014/8

N2 - We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (α, α2) or (α, α - α2) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u, u′) with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u, u′) has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

AB - We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (α, α2) or (α, α - α2) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u, u′) with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u, u′) has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

KW - Cubic number fields

KW - Hermite problem

KW - Multidimensional continued fractions

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U2 - 10.1007/s00605-014-0643-1

DO - 10.1007/s00605-014-0643-1

M3 - Article

AN - SCOPUS:84905025291

VL - 174

SP - 549

EP - 566

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 4

ER -