# Tagged Questions

**0**

votes

**0**answers

56 views

### Are major arcs always around a fraction with small denominator? [on hold]

In the usual circle method we might have a trigonometric polynomial $F(\theta)=\sum_{n}a_n e(n\theta)$ and we need to estimate the integral $\int_0^1 F(\theta)d\theta$ by breaking the domain into ...

**11**

votes

**0**answers

350 views

### Correlation of Fourier transforms of characteristic functions

Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...

**12**

votes

**1**answer

629 views

### On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of ...

**5**

votes

**1**answer

373 views

### Fourier analysis, orthogonality, and Plancherel for finite abelian groups

I am reading an outstanding paper by Bateman and Katz, improving the best known bounds on the cap set problem (Roth's theorem over $\mathbb{F}_3^N$).
The paper contains some technical lemmas for ...

**3**

votes

**2**answers

827 views

### Choice of normalization of the finite Fourier transform

Let $f : \mathbb{Z}/N\mathbb{Z} \rightarrow \mathbb{C}$ be a function. Is its Fourier transform typically defined by
$$\widehat{f}(\xi) = \sum_{x \in \mathbb{Z}/N \mathbb{Z}} f(x) e^{- 2 \pi i x \xi ...

**15**

votes

**3**answers

1k views

### What is the shortest route to Roth's theorem?

Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the ...

**6**

votes

**1**answer

803 views

### Additive combinatorics and large Fourier coefficients

Elon Lindenstrauss explains in his talk at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $\mu$ on the ...