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
SN - 0026-9255
VL - 174
SP - 549
EP - 566
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
IS - 4
ER -