# Tagged Questions

**0**

votes

**0**answers

230 views

### What is $p$-adic Fourier series?

Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$?
Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion:
...

**4**

votes

**1**answer

181 views

### sum of integral part of n/k

Is there any direct formula or algorithm better than the brute force (O(n) algorithm by iterating from 1 to n) way to calculate the sum
\begin{equation}
S = \sum\limits_{i=1}^n [{\frac{n}{i}}]
...

**4**

votes

**1**answer

714 views

### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...

**2**

votes

**3**answers

195 views

### Speed of convergence for Weyl's Equidistribution theorem

If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...

**3**

votes

**0**answers

145 views

### The polynomial Freiman-Ruzsa conjecture for the special case when $f$ is a bijection

The polynomial Freiman-Ruzsa conjecture states that there exists $k > 0$, such that for all $\epsilon > 0$, and for all large $n$ and all functions $ f: \mathbb{F}_2^n \mapsto\mathbb{F}_2^n$, ...

**1**

vote

**0**answers

81 views

### Dependence between $\langle f(x), g(y) \rangle$ and $\langle x, y \rangle$.

Let $p$ be a large prime and $n \geq 3$ be an integer. Let $f$ and $g$ be two arbitrary bijections from $\mathbb{F}_p^n$ to $\mathbb{F}_p^n$. I want to find a condition on $f$ and $g$ such that the ...

**3**

votes

**1**answer

403 views

### Uncertainty principle (really for Mellin, but never mind that!)

Is there a smooth funtion $f:\mathbb{R}\to \mathbb{C}$ such that
(a) $f(x)$ decreases faster than $e^{-e^x}$ when $x\to \infty$,
(b) $\widehat{f}(t)$ decreases faster than $e^{-|t|}$ when $t\to ...

**4**

votes

**0**answers

155 views

### Fourier analysis on crystallographic groups

It seems pretty clear a priori that Fourier analysis on a discrete cocompact group of motions of $\mathbb{E}^n$ should be quite tractable (since, by Bieberbach, such groups are almost $\mathbb{Z}^n,$ ...

**4**

votes

**1**answer

485 views

### Max of Fourier transform?

Let f be a real-valued function (or distribution) on $\mathbb{R}$. (You can assume it is nice in one way or another.) What would be some practical ways to bound
$\max_{\alpha \in \mathbb{R}} ...

**1**

vote

**1**answer

297 views

### A question about the quotient measure on the ideles and the adeles

Let $F$ be a local field, then the additive Haar measure is easy to relate to the multiplicative Haar measure:
$$ \int_F f(x) d^+ x = \int_{F^\times} f(x) |x| d^+ x.$$
I know that the ideles have ...

**2**

votes

**3**answers

862 views

### Automorphic Forms on product of groups $G\times H$

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.
Let $G$, $H$ be two reductive groups defined over a number field
$F$. Let $\mathcal{A}(G)$ be ...

**1**

vote

**1**answer

156 views

### Local nonarchimedean Sobolev inequality

Let $B$ be the unit ball in $\mathbb{R}^n$. A local version of the Sobolev inequality on $\mathbb{R}^n$ says that for any $p\in[1,\infty]$ there exist constants $C >0$ and $k \in \mathbb{N}$ such ...

**6**

votes

**1**answer

570 views

### Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces?

Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. ...

**3**

votes

**2**answers

259 views

### Traceable representation of reductive group over a p-Adic field.

I have a question on how to define a trace of a unitary representation. If $G$ is a reductive group over a p-adic field it is known ( I do not know who prove it) that is of type I. Knowing this we ...

**5**

votes

**5**answers

2k views

### Exponential sums for beginner.

What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...

**7**

votes

**3**answers

920 views

### A Question concerning the Fourier Transform of $\mathbb{R}$

Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...