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age 26
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PhD student at Princeton University


Apr
10
comment Reference for Kronecker-Weyl theorem in full generality
The comment I made to your other answer also applies here: these only deal with the discrete form of Kronecker-Weyl in the case of linear independence.
Apr
10
comment Reference for Kronecker-Weyl theorem in full generality
Again, this does not answer the question as it deals only with the case of linear independence (and also it is about the discrete form of the Kronecker-Weyl theorem, not the continuous form).
Apr
9
comment Reference for Kronecker-Weyl theorem in full generality
@GregMartin - done!
Apr
9
comment Reference for Kronecker-Weyl theorem in full generality
Oh, I think I get it; you mean a base change such that $\theta_1,\ldots,\theta_r$ are linearly independent over $\mathbb{Q}$ and $\theta_{r+1},\ldots,\theta_n = 0$. But this base change is essentially equivalent to part (1) of the theorem in my answer, and then at the end for part (2) you're going to need to apply the inverse of this base change to get the result.
Apr
9
answered Reference for Kronecker-Weyl theorem in full generality
Apr
9
comment Reference for Kronecker-Weyl theorem in full generality
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
comment Reference for Kronecker-Weyl theorem in full generality
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
comment Reference for Kronecker-Weyl theorem in full generality
This is incorrect; the theorem depends on the linear relations between the $\theta_j$, because this defines the subtorus!
Apr
9
comment Reference for Kronecker-Weyl theorem in full generality
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
2
revised A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights
added 269 characters in body
Apr
2
revised A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights
added 247 characters in body
Apr
2
revised A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights
added 1714 characters in body
Apr
2
answered A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights
Mar
7
awarded  Popular Question
Feb
5
awarded  Yearling
Dec
5
awarded  Good Question
Dec
2
asked Adelic methods for classical modular forms
Oct
3
comment Zeros of Ramanujan tau L-function
Try Analytic Number Theory by Iwaniec and Kowalski. I imagine it's in Chapter 5.
Oct
2
comment Backlund counting formula for Dirichlet L-functions?
Your alleged proof does not include GRH as a corollary; rather, your claim is that the number of zeroes off the line is $o(T\log T)$. This, however, would nevertheless be a huge breakthrough result, as for example for $\zeta(s)$ it is only currently known that $\limsup_{T\to\infty}N_0(T)/N(T)\geq 0.42$, or heuristically that "at least 42% of the zeroes of $\zeta(s)$ lie on the critical line", whereas you claim 100%. From a cursory glance of your paper, however, I have my doubts as there seems to be no "heavy lifting" (though it is hard to read as the notation is often nonstandard).
Jul
30
comment Distribution of integers with number of prime factors lying in a given arithmetic progression
See also the following paper where the size of the error term is also discussed: projecteuclid.org/euclid.nmj/1306851587