A solution to a problem of Cassels and Diophantine properties of cubic numbers
| dc.creator | Shapira, Uri | |
| dc.date | 2008-10-23 | |
| dc.date | 2009-05-07 | |
| dc.date.accessioned | 2026-07-07T13:12:01Z | |
| dc.date.available | 2026-07-07T13:12:01Z | |
| dc.description | We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n|\to\infty} |n|<na - x> <nb - y> = 0, where <-> denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type. | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/0810.4289 | |
| dc.identifier | http://arxiv.org/abs/0810.4289 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/229465 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.title | A solution to a problem of Cassels and Diophantine properties of cubic numbers | |
| dc.type | text |