All Questions
14 questions
1
vote
0
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127
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an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
8
votes
0
answers
240
views
Question on calculating character sums
I am wondering if there are any references that would help me with the following problem:
Let $p>2$ be a prime number, $n \in \mathbb{Z}^+$ odd such that $(n,p)=1$, $\chi$ an imaginary quadratic ...
- 707
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
8
votes
2
answers
1k
views
Lower bound on exponential sums
Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
https://...
- 751
4
votes
1
answer
208
views
Stationary phase method for $\varphi''(x_0)= 0$
Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...
- 12.3k
4
votes
0
answers
200
views
When does a continuous function's "Fourier series" converge pointwise almost everywhere to the function?
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
- 6,180
4
votes
1
answer
383
views
A "nice" trigonometric polynomial approximation of a characteristic function
Let $\delta > 0$ be small and $\chi_{[-\delta, \delta]}(t)$ be a characteristic function on the interval $[-\delta, \delta]$. I am interested in a trigonometric polynomial $S$ such that
$$
\chi_{[-\...
- 3,625
1
vote
1
answer
130
views
distance-set along the orbit of $e^{2\pi i\theta}$
Let $z=e^{2\pi i\theta}$ for a fixed real number $\theta$. It's known that if $\theta\not\in\mathbb{Q}$ (is irrational) then the set $S(\theta)=\{z^n: n\in\mathbb{N}\}$ is dense on the unit circle $\...
- 43.2k
9
votes
1
answer
646
views
Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?
In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...
- 2,901
1
vote
0
answers
285
views
Davenport's proof that almost all integers are the sum of 4 cubes
Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
- 1,890
14
votes
4
answers
3k
views
Fourier decay rate of Cantor measures
For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
- 981
6
votes
2
answers
861
views
Number of integers coprime to l
A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for
$$
\sum_{n \leq x, (n, \ell) = 1} 1
$$
Of course, this is easy to estimate with a trivial error term of $O(\varphi(l))...
- 293
6
votes
0
answers
252
views
How big is the Fourier transform of the log of a polynomial over the p-adic numbers
Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that ...
- 527
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795