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Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms
| Content Provider | Semantic Scholar |
|---|---|
| Author | Dasaratha, Krishna Flapan, Laure Garrity, Thomas Lee, Chan-Soo Mihaila, Cornelia Neumann-Chun, Nicholas Peluse, Sarah Stoffregen, Matthew |
| Copyright Year | 2012 |
| Abstract | 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 $$\alpha $$α so that either $$(\alpha , \alpha ^2)$$(α,α2) or $$(\alpha , \alpha -\alpha ^2)$$(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 $$(u, u')$$(u,u′) with $$u$$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 $$(u, u')$$(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. |
| Starting Page | 549 |
| Ending Page | 566 |
| Page Count | 18 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00605-014-0643-1 |
| Volume Number | 174 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1208.4244.pdf |
| Alternate Webpage(s) | https://arxiv.org/pdf/1208.4244v5.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/1208.4244.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s00605-014-0643-1 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
