Timeline for Reference for Kronecker-Weyl theorem in full generality
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 22, 2020 at 8:17 | comment | added | Peter Humphries | Lemma 7 of doi.org/10.1112/S0024611504014741 calls it "well-known". | |
| Oct 21, 2020 at 23:51 | answer | added | Günter Rote | timeline score: 2 | |
| Oct 21, 2020 at 18:56 | comment | added | Günter Rote | Can you give a reference where the theorem is stated, and called "classical" and "well-known"? | |
| Oct 21, 2020 at 18:54 | answer | added | Günter Rote | timeline score: 2 | |
| Apr 15, 2014 at 19:01 | vote | accept | Greg Martin | ||
| Apr 10, 2014 at 5:23 | comment | added | Marc Palm | Sorry, I didn't see the part: " a proof under the assumption that the θj are linearly independent over the rational numbers will not suffice for me." I deleted my answer. | |
| Apr 9, 2014 at 20:40 | comment | added | Peter Humphries | @GregMartin - done! | |
| Apr 9, 2014 at 20:33 | comment | added | Greg Martin | @PeterHumphries, would you be willing to send me a copy of your thesis? I appreciate the details you supplied below, but I expect I'd get even more out of your write-up and bibliography. | |
| Apr 9, 2014 at 18:59 | answer | added | Peter Humphries | timeline score: 19 | |
| Apr 9, 2014 at 18:18 | comment | added | Peter Humphries | Asaf, did you check my reference? Witte Morris' book discusses the Kronecker-Weyl theorem directly in the first three sections of the first chapter (as a motivating case of Ratner's theorems), which is what Greg Martin was asking about. I didn't say that Witte Morris' book actually gave the complete proof of Ratner's theorems. In any case, the proof of the Kronecker-Weyl theorem is really a theorem of abelian Fourier analysis, so even the nilpotent case of Ratner's theorems is overkill. | |
| Apr 9, 2014 at 18:09 | comment | added | Asaf | @Peter, first, one would need to adapt Ratner's proof for the discrete case, this certainly can be done, but it's more delicate issue than the semi-simple case, where just by disjointness you conclude that in two lines. Ratner's can be indeed proven rather easily in the nilpotent case (see for example in the upcoming Eisendler-Ward book (vol III), Witte-Moris' book's proof is far from complete, except in the $SL_{2}$ case), but usually proofs regarding nilpotent groups are dealt directly with the use of Malcev theory (i.e. in Raghunathan's book), where Kronecker's lemma is used extensively. | |
| Apr 9, 2014 at 17:22 | comment | added | Peter Humphries | For Ratner's theorem, you could use this as a reference: people.uleth.ca/~dave.morris/books/Ratner.pdf. It covers the basic case of $G/\Gamma = \mathbb{R}^d/\mathbb{Z}^d$ in the beginning of the first chapter, but unfortunately it leaves the proofs as exercises, namely Exercises 1.1.2, 1.1.5, and 1.3.4. | |
| Apr 9, 2014 at 15:35 | comment | added | Asaf | Another place to get your theorem is through Furstenberg's proof of the density/equidistribution theorem (depends on your favourite category of dynamics), as that the statement for extensions of minimal systems (topological dynamics) would conclude that the resulting extension is semi-simple (that is, union of tori). | |
| Apr 9, 2014 at 15:34 | comment | added | Asaf | One way to get that would be to use Ratner's theorem (the topological one), which is stated as you would like, but that seems like an overkill to me, as you just dealing with a nilflow. If I recall correctly, such a statement as you wish (and even in a quantitative manner) appears in one of the Green-Tao papers (and therefore, the purely qualitative theorem is probably appearing in one of Berglson's papers). | |
| Apr 9, 2014 at 14:50 | comment | added | Peter Humphries | If you just wish for a complete proof, I can send you a copy of my honours thesis, where I prove the result from scratch. But I doubt that'd suffice for a reference should you wish for a citation. That being said, the reason that I wrote out the proof in full is that at the time (2010) I couldn't find a reference where the result is proved for the case where the $\theta_j$ may not be linearly independent. | |
| Apr 9, 2014 at 12:57 | comment | added | Gerry Myerson | First place I'd look would be the Kuipers and Niederreiter book, though I don't know whether it's in there in the form you want. | |
| Apr 9, 2014 at 9:31 | answer | added | MHMertens | timeline score: 1 | |
| Apr 9, 2014 at 7:58 | history | asked | Greg Martin | CC BY-SA 3.0 |