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PhD student at Princeton University


1d
reviewed Approve suggested edit on $K$-homology of $BG$
Jul
12
reviewed Approve suggested edit on Incomplete Kloosterman sum
Jul
9
awarded  Popular Question
Jun
28
revised Dedekind Zeta function: behaviour at 1
fixed link
Jun
26
reviewed Approve suggested edit on How do you show that $S^{\infty}$ is contractible?
Jun
18
reviewed Reject suggested edit on nontrivial theorems with trivial proofs
Jun
4
awarded  Nice Question
May
28
awarded  Custodian
May
28
reviewed Approve suggested edit on The prime numbers modulo $k$, are not periodic
May
25
awarded  Enlightened
May
25
awarded  Nice Answer
Apr
24
comment Petersson product of newforms to different level
What do you mean by the Petersson inner product? By definition it should be of the form $\langle f,g\rangle = \frac{1}{[\Gamma_0(1) : \Gamma_0(N)]}\int_{\Gamma_0(N) \backslash \mathbb{H}}f(z)\overline{g(z)} y^k \, \frac{dx\, dy}{y^2}$, and in particular depends on $N$ (and $k$). So it only makes sense to take the inner product of two modular forms of the same weight and level.
Apr
22
answered Prime races à la Mertens
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!