### 数学代写|傅里叶分析代写Fourier analysis代考|AMTH246

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|傅里叶分析代写Fourier analysis代考|A First Sketch of the Argument

We start by recalling, very briefly, the usual approach taken in proving Roth’s Theorem. One takes a set $A$ of density $\delta$ in $\mathbb{Z} / N \mathbb{Z}$, and compares the number of length 3 Arithmetic Progressions in $A$ with $\frac{1}{2} \delta^3 N^2$. This is roughly the number of 3-term APs in a random subset of $\mathbb{Z} / N \mathbb{Z}$. The difference $D$ between these two quantities can be expressed using the Fourier Coefficients $\hat{A}(r)$ of $A$. If $D$ is small then $A$ contains a progression of length 3 becuase it approximates a random set. Otherwise $D$ is large, and we can deduce that some $\hat{A}(r)$ is large for $r \neq 0$. This information in turn allows us to deduce that $A$ has increased density $\delta+c \delta^2$ in some reasonably large Arithmetic Progression $P$. But $P$ is affinely equivalent to ${1, \ldots, N}$, and so we can iterate the argument. However one can only increment the density $O\left(\delta^{-1}\right)$ times before it becomes greater than 1, which is clearly impossible. Hence if $A$ is large enough then it contains a 3 -term AP.
Bourgain’s point of departure seems to be the following. Suppose that
$$\hat{A}(r)=\sum_n A(n) e^{2 \pi i n r / N}$$
is large. To show that $A$ has increased density in some progression $P$, one has to somehow get rid of the exponential terms appearing here. In the usual proof of Roth’s Theorem this is done by splitting up $\mathbb{Z} / N \mathbb{Z}$ into small progressions on which $e^{2 \pi i n r / N}$ is roughly constant as $n$ varies. This, however, is rather inefficient – rather a lot of small progressions are required. Suppose instead that one forgets about progressions, and splits $\mathbb{Z} / N \mathbb{Z}$ up into sets on which $|n r / N|$ is roughly constant. We could easily deduce that $A$ has increased density on one of these sets. Unfortunately however this information is not equivalent to the original hypothesis, since one of the new sets is not affinely equivalent to ${1, \ldots, N}$. Hence we have to strengthen the entire hypothesis that we are trying to prove.

The “sets” that we are discussing here are of course just translates of Bohr Neighbourhoods. Hence we shall try to prove something like the following.

Conjecture 4 Let $A$ be a subset of some Bohr Neighbourhood $\Lambda$, such that $|A|=\delta|\Lambda|$. Then for fixed $\delta$ and “sufficiently large” $\Lambda, A$ contains a three-term Arithmetic Progression.

Since $\mathbb{Z} / N \mathbb{Z}$ is trivially a Bohr Neighbourhood, we might hope that this would imply Roth’s Theorem with a better bound.

There are many difficulties to overcome in order to make the above idea work, as we shall discover. These stem principally from three facts.

## 数学代写|傅里叶分析代写Fourier analysis代考|Definitions and Elementary Properties

We begin by defining what we mean by a Bohr Neighbourhood from now on.
Definition 5 Let $\theta=\left{\theta_1, \ldots, \theta_d\right} \in \mathbb{R}^d$, and let $\epsilon$ and $M$ be real numbers with $\epsilon<\frac{1}{2}$. Then we define the Bohr Neighbourhood $\Lambda_{\theta, \epsilon, M}$ to be the set of all $n \in \mathbb{Z}$ such that $|n| \leq M$ and $\left|n \theta_j\right| \leq \epsilon$ for $j=1, \ldots, d$.

This is clearly very similar to the “mod $N$ ” version of the same name. We take the opportunity to record here some simple facts about Bohr Neighbourhoods which will be useful later.
Lemma $6\left|\Lambda_{\theta, \epsilon, M}\right| \geq \epsilon^d M$
Proof Let $\mathbb{S}^d$ be the unit torus $\mathbb{R}^d / \mathbb{Z}^d$. Consider the set of all $P_n=\left(\left|n \theta_1\right|, \ldots,\left|n \theta_d\right|\right) \in \mathbb{S}^d$ for integers $n \in[1, M]$. This has size $M$, so some $\epsilon$-cube $\mathcal{B}$ of $\mathbb{S}^d$ contains at least $M \epsilon^d$ of the $P_i$ (this “obvious” averaging argument actually requires careful analysis its justification). Let $\mathcal{C}$ be the set of all $n \in[1, M]$ for which $P_n \in \mathcal{B}$. Then there is an injection
$$\phi: \mathcal{C} \rightarrow \Lambda_{\theta, \epsilon, M}$$
defined by $\phi(n)=n-n_0$, where $n_0 \in \mathcal{C}$ is arbitrary.
Lemma $7\left|\Lambda_{\theta, \epsilon, M}\right|<8^{d+1}\left|\Lambda_{\theta, \frac{\epsilon}{2}, \frac{M}{2}}\right|$
Proof Divide $\Lambda_{\theta, \epsilon, M}$ into sets $A_i$ such that
(i) $\left{\left(\left|n \theta_1\right|, \ldots,\left|n \theta_d\right|\right) \mid n \in A_i\right}$ is contained in an $\frac{\epsilon}{2}$-cube in $\mathbb{S}^d$;
(ii) $A_i$ is contained in an interval of length $\frac{M}{2}$.
This can be achieved with $8^{d+1}$ sets $A_i$. Each $A_i$ injects to $\Lambda_{\theta, \frac{5}{2}, \frac{M}{2}}$ by sending $n$ to $n-n_0$, where $n_0 \in A_i$ is arbitrary. The result follows.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|A First Sketch of the Argument

Bourgain 的出发点似乎是以下几点。假设
$$\hat{A}(r)=\sum_n A(n) e^{2 \pi i n r / N}$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Definitions and Elementary Properties

$$\phi: \mathcal{C} \rightarrow \Lambda_{\theta, \epsilon, M}$$

(二) $A_i$ 包含在长度区间内 $\frac{M}{2}$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。