### 分类： 加性组合代写

• 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

Topological Multiple Recurrence. Let $X$ be a compact metric space, and $T$ be a continuous map. For any integer $k \geq 1$ there exists a point $x \in X$ and a sequence $n_{\ell} \rightarrow \infty$ with $T^{j n_{\ell}} x \rightarrow x$ for each $1 \leq j \leq k$.

This theorem is analogous to van der Waerden’s theorem, and indeed implies it. To see this, let $\Lambda={1, \ldots, r}$ represent $r$ colors, and consider $\Omega=\Lambda^Z$. Thus $\Omega$ is the space of all $r$ colorings of the integers, and by $x \in \Omega$ we understand a particular $r$ coloring of the integers. We make $\Omega$ into a compact metric space (check using sequential compactness), by taking as the metric $d(x, y)=0$ if $x=y$ and $d(x, y)=2^{-\ell}$ where $\ell$ is the least magnitude for which either $x(\ell) \neq y(\ell)$ or $x(-\ell) \neq y(-\ell)$. We define the shift map $T$ by $T x(n)=x(n+1)$. Now suppose we are given a coloring $\xi$ of the integers. Take $X$ to be the closure of $T^n \xi$ where $n$ ranges over all integers. By definition this is a closed invariant compact metric space, and so by the Topological Multiple Recurrence Theorem there is a $x \in X$ and some $n \in \mathbb{Z}$ with $x(0)=x(n)=x(2 n)=\ldots=x(k n)$. But from the definition of the space $X$ we may find an $m \in \mathbb{Z}$ such that $T^m \xi$ and $x$ agree on the interval $[-k n, k n]$. Then it follows that $\xi(m)=\xi(m+n)=\ldots=\xi(m+k n)$ producing a $k+1$ term AP.
The above argument gives an infinitary version of the van der Waerden theorem where we color all the integers. But from it we may deduce the finite version. Suppose not, and there are $r$ colorings of $[-N, N]$ with no monochromatic $k$-APs for each natural number $N$. Extend each of these colorings arbitrarily to $\mathbb{Z}$, obtaining an element in $\Omega$. By compactness we may find a limit point in $\Omega$ of these elements. That limit point defines a coloring of $\mathbb{Z}$ containing no monochromatic $k$-APs, and this is a contradiction.

The ergodic theoretic analog of Szemerédi’s theorem is Furstenberg’s multiple recurrence theorem for measure preserving transformations, and this implies Szemerédi by an argument similar to the one above.

Furstenberg’s Theorem. Let $X$ be a probability measure space and let $T$ be a measure preserving transformation. If $V$ is a set of positive measure, then there exists a natural number $n$ such that $V \cap T^{-n} V \cap T^{-2 n} V \cap \ldots \cap T^{-k n} V$ has positive measure.

We begin with a warm-up result, which although unrelated may help set the mood.
Schur’s Theorem. Given any positive number $r$, if $N \geq N(r)$ and the integers in $[1, N]$ are colored using $r$ colors then there is a monochromatic solution to $x+y=z$.
First we need a special case of Ramsey’s theorem.
Lemma. Suppose that the edges of the complete graph $K_N$ are colored using $r$ colors. If $N \geq N(r)$ then there is a monochromatic triangle.

Proof. We will use induction on $r$. It is very well known that if $r=2$ and $N \geq 6$ then there is a monochromatic triangle. Suppose we know the result for $r-1$ colorings, and we need $N \geq N(r-1)$ for that result. Pick a vertex. There are $N-1$ edges coming out of it. So for some color there are $\geq\lceil(N-1) / r\rceil$ edges starting from this vertex having the same color. Now the complete graph on the other vertices of these edges must be colored using only $r-1$ colors. Thus if $N \geq r N(r-1)-r+2$ we are done.

Proof of Schur’s Theorem. Consider the complete graph on $N$ vertices labeled 1 through $N$. Color the edge joining $a$ to $b$ using the color of $|a-b|$. By our lemma, if $N$ is large then there is a monochromatic triangle. Suppose its vertices are $a<b<c$ then $(c-a)=(c-b)+(b-a)$ is a solution proving Schur’s theorem.

Let $k$ and $r$ be given natural numbers. Consider the cube $[1, k]^N$, and color each point in it using $r$ colors. The Hales-Jewett theorem says that if $N$ is sufficiently large then there will be a monochromatic line having $k$ points. Here a (combinatorial) line means the following: Let $\mathrm{x}=\left(x_1, \ldots, x_N\right)$ be a point, and let $A$ be a non-empty subset of $[1, N]$. By $\mathbf{x} \oplus j A$ (where $1 \leq j \leq k$ ) we denote the point $\mathbf{y}(j$ ) whose coordinates are given by $y_i(j)=x_i$ if $i \notin A$ and $y_i(j)=j$ if $i \in A$. The line $\mathbf{x} \oplus A$ consists of the points $\mathbf{x} \oplus j A$ for $1 \leq j \leq k$. In other words, $A$ describes a set of coordinates whose entries are wildcards taking all the values from 1 to $k$.

As a special case consider $k=3$ and $r=2$ which corresponds (essentially) to a game of tic-tac-toe. The Hales-Jewett theorem guarantees that in high dimension a game of tic-tac-toe never ends in a draw. Since the first person has a free move, and can steal any winning strategy that the second person devises, it follows that the first player should win such games.

We will now give two proofs of the Hales-Jewett theorem; the second, due to Shelah, being a small but very important modification of the first. The proofs both proceed by induction on $k$ and $r$. Let $H J(k, r)$ denote the least $N$ for which the theorem holds; we wish to show that this is finite, and also derive some bounds for it. Note that if $k=1$ there is nothing to prove and we may take $H J(1, r)=1$. Consider next the case that $k=2$. Take $N=r$ and note that two of the $r+1$ points $(1,1, \ldots, 1),(1,1, \ldots, 1,2),(1, \ldots, 2,2)$, $\ldots,(1,2,2, \ldots, 2),(2,2, \ldots, 2)$ must have the same color. Thus $H J(2, r) \leq r$. Exercise: show that $H J(2, r)=r$.

# 加性组合代写

Szemerédi 定理的遍历理论类比是 Furstenberg 的保测变换的多重递归定理，这通过与上述类似的论证 暗示了 Szemerédi。

Furstenberg 定理。让 $X$ 是一个概率测度空间，让 $T$ 是一种保量变换。如果 $V$ 是一组正测度，则存在一个 自然数 $n$ 这样 $V \cap T^{-n} V \cap T^{-2 n} V \cap \ldots \cap T^{-k n} V$ 有积极的措施。

## 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。

• 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

Gowers’s Theorem. There exists a positive constant $c_k$ such that any subset $A$ in $[1, N]$ with $|A| \gg N /(\log \log N)^{c_k}$ contains a non-trivial $k$ term arithmetic progression.

In this course, we hope to give an account of Gowers’s proof in the case $k=4$. One of the major insights of Gowers is the development of a “quadratic theory of Fourier analysis” which substitutes for the “linear Fourier analysis” used in Roth’s theorem. Gowers’s ideas have transformed the field, opening the door to many spectacular results, most notably the work of Green and Tao.

The Green-Tao Theorem (2003). The primes contain arbitrarily long non-trivial arithmetic progressions.

Note that up to $N$ there are about $N / \log N$ primes. This density is much smaller than what would be covered by Gowers’s theorem; even in the case $k=3$ it is not covered by the best known results on $r_3(N)$. We will not be able to cover the Green-Tao theorem, but will give some of the ideas in the simple case $k=3$. Another result along those lines is the celebrated three primes theorem.

Vinogradov’s theorem (1937). Every large odd number is the sum of three primes.
Another brilliant result of Green and Tao, developing Gowers’s ideas, is that $r_4(N) \ll$ $N(\log N)^{-c}$ where $r_4(N)$ denotes the largest cardinality of a set in $[1, N]$ containing no four term progressions.

Another theme that we shall explore, and which also plays an important role in Gowers’s proof, is Freiman’s theorem on sumsets. If $A$ is a set of $N$ integers then $A+A$ is bounded above by $N(N+1) / 2$, and below by $2 N-1$. The lower bound is attained only when $A$ is highly structured, and is an arithmetic progression of length $N$. Clearly if $A$ is a subset of an arithmetic progression of length $C N$ then $|A+A| \leq 2 C|A|$. More generally suppose $d_1, \ldots, d_k$ are given numbers, and consider the set
$$\left{a_0+a_1 d_1+\ldots+a_k d_k: \quad 1 \leq a_i \leq N_i \text { for } 1 \leq i \leq k\right}$$
We may think of this as a generalized arithmetic progression of dimension $d$. Note that this generalized AP has cardinality at most $N_1 \cdots N_k$. If these sums are all distinct (so that the cardinality equals $N_1 \cdots N_k$ ) we call the GAP proper. Note that if $A$ is contained in a gAP of dimension $k$ and size $\leq C N$ then $|A+A| \leq 2^k C N$. Freiman’s theorem provides a converse to this showing that all sets with small sumsets must arise in this fashion.
Freiman’s theorem. If $A$ is a set with $|A+A| \leq C|A|$ then there exists a proper GAP of dimension $k$ (bounded in terms of $C$ ) and size $\leq C_1|A|$ for some constant $C_1$ depending only on $C$.

Qualitatively Freiman’s theorem says that any set with a small sumset looks like an arithmetic progression. Similarly we may expect that a set with a small product set should look like a geometric progression. But of course no set looks simultaneously like an arithmetic and a geometric progression! Thus we may surmise, as did Erdős and Szemerédi that either the sumset or the product set must be large.

This is currently known for $\epsilon>3 / 4$ (indeed a little better) thanks to results of ErdősSzemerédi, Solymosi, Elekes … . The sum-product theory (and its generalizations) is another very active problem in additive combinatorics, and has led to many important applications (bounding exponential sums etc).

We end this introduction by giving a brief description of how ergodic theory connects up with these combinatorial problems. The subject begins with a simple recurrence theorem of Poincaré.

Poincaré recurrence. Let $X$ be a probability space with measure $\mu$, and let $T$ be a measure preserving transformation (so $\mu\left(T^{-1} A\right)=\mu(A)$ ). For any set $V$ with positive measure there exists a point $x \in V$ such that for some natural number $n, T^n x$ also is in $V$.

Proof. This is very simple: note that the sets $V, T^{-1} V, T^{-2} V, \ldots$ cannot all be disjoint. Therefore $T^{-m} V \cap T^{-m-n} V \neq \emptyset$ for some natural numbers $m$ and $n$. But this gives readily that $V \cap T^n V \neq \emptyset$ as needed.

It is clear from the proof that the number $n$ in Poincaré’s result may be found below $1 / \mu(V)$. As an example, we may take $X$ to be the circle $\mathbb{R} / \mathbb{Z}$, and take $V$ to be the interval $[-1 / 2 Q, 1 / 2 Q]$, and $T$ to be the map $x \rightarrow x+\theta$ for some fixed number $\theta$. We thus obtain:
Dirichlet’s Theorem. For any real number $\theta$, and any $Q \geq 1$ there exists $1 \leq q \leq Q$ such that $|q \theta| \leq 1 / Q$. Here $|x|$ denotes the distance between $x$ and its nearest integer.
If $X$ happens also to be a separable (covered by countably many open sets) metric space, then we can divide $X$ into countably many balls of radius $\epsilon / 2$. Then it follows that almost every points of $X$ returns to within $\epsilon$ of itself. That is, almost every point is recurrent.
We don’t really need a probability space to find recurrence. Birkhoff realized that this can be achieved purely topologically and holds for compact metric spaces.
Birkhoff’s Recurrence Theorem. Let $X$ be a compact metric space, and $T$ be a continuous map. Then there exists a recurrent point in $X$; namely, a point $x$ such that there is a sequence $n_k \rightarrow \infty$ with $T^{n_k} x \rightarrow x$.

Proof. Since $X$ is compact, any nested sequence of non-empty closed sets $Y_1 \supset Y_2 \supset Y_3 \ldots$ has a non-empty intersection. Consider $T$-invariant closed subsets of $X$; that is, $Y$ with $T Y \subset Y$. By Zorn’s lemma and our observation above, there exists a non-empty minimal closed invariant set $Y$. Let $y$ be any point in $Y$ and consider the closure of $y, T y, T^2 y, \ldots$. This set is plainly a closed invariant subset of $Y$, and by minimality equals $Y$. Therefore $y$ is recurrent.

These are some basic simple results, of the same depth as Dirichlet’s pigeonhole principle and its application to Diophantine approximation. In the example of Diophantine approximation, we see that if $|n \theta|$ is small then so are $|2 n \theta|,|3 n \theta|$ etc. This suggests the possibility of multiple recurrence.

# 加性组合代写

Green-Tao 定理 (2003)。素数包含任意长的非平凡算术级数。

Veft{a_0+a_1 d_1+\ldots+a_k d_k: \quad 1 \eq a_i \leq N_i \text { for } 1 Veq i \eq k\right } }

## 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。

• 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

The aim of this course is to study additive problems in number theory. Broadly, given a sufficiently large set of integers $A$ (or more generally a subset of some abelian group) we are interested in understanding additive patterns that appear in $A$. An important example is whether $A$ contains non-trivial arithmetic progressions of some given length $k$. One reason for considering arithmetic progressions is that they are quite indestructible structures: they are preserved under translations and dilations of $A$, and they cannot be excluded for trivial congruence reasons. For example the pattern $a, b$ and $a+b$ all being in the set seems quite close the arithmetic progression case $a, b,(a+b) / 2$, but the former case can never occur in any subset of the odd integers (and such subsets can be very large). Another class of questions we can ask is whether all numbers can be written as a sum of $s$ elements from a given set $A$. For example, all numbers are sums of four squares, nine cubes etc. Waring’s problem and the Goldbach conjectures are two classical examples. In the same spirit, given a set $A$ of $N$ integers we may ask for information about the sumset $A+A:={a+b: a, b \in A}$. If there are not too many coincidences, then we may expect $|A+A| \gg N^2$. But when $A$ is an AP note that $|A+A| \leq 2|A|-1$. One of our goals for the class will be Freiman’s theorem that if the sumset is small then $A$ looks like a “generalized arithmetic progression.”

The subject may be said to begin with a beautiful result of van der Waerden (1927).
van der Waerden’s Theorem. Let $k$ and $r$ be given. There exists a number $N=N(k, r)$ such that if the integers in $[1, N]$ are colored using $r$ colors, then there is a non-trivial monochromatic $k$ term arithmetic progression.
van der Waerden’s proof was by an ingenious elementary induction argument on $k$ and $r$. The proof does not give any good bound on how large $N(k, r)$ needs to be. A more general result was subsequently found by Hales and Jewett (1963), with a nice refinement of Shelah (1988), but again the bounds for the van der Waerden numbers are quite poor.
The Hales-Jewett Theorem. Let $k$ and $r$ be given. There exists a number $N=N(k, r)$ such that if the points in $[1, k]^N$ are colored using $r$ colors then there is a monochromatic “combinatorial line”. Here a combinatorial line is a collection of $k$ points of the following type: certain of the coordinates are fixed, and a certain non-empty set of coordinates are designated as “wildcards” taking all the values from 1 to $k$.

A picturesque way of describing the Hales-Jewett theorem is that a “tic-tac-toe” game of getting $k$ in a row, played by $r$ players, always has a result in sufficiently high dimensions. Since there is obviously no disadvantage to going first, the first player wins; but no constructive strategy solving the game is known. One can recover van der Waerden’s theorem by thinking of $[1, k]^N$ as giving the base $k$ digits (shifted by 1) of numbers in $\left[0, k^N-1\right]$

Erdős and Turan proposed a stronger form of the van der Waerden, partly in the hope that the solution to the stronger problem would lead to a better version of van der Waerden’s theorem.

In other words, $N(\delta, 3) \leq \exp (\exp (C / \delta))$ for some positive constant $C$. This stronger result does in fact give a good bound on the van der Waerden numbers for $k=3$. We know now thanks to Bourgain that $|A| \gg N(\log \log N / \log N)^{1 / 2}$ suffices. Thus the double exponential bound can be replaced by a single exponential.

Let $r_3(N)$ denote the size of the largest subset of $[1, N]$ having no non-trivial three term APs. Then as mentioned above, $r_3(N) \ll N \sqrt{\log \log N / \log N}$. What is the true nature of $r_3(N)$ ? If we pick a random set $A$ in $[1, N]$ we may expect that it has about $|A|^3 / N$ three term APs. This suggests that $r_3(N)$ is perhaps of size $N^{1 / 3}$. However, in 1946 Behrend found an ingenious construction that does much much better.

Behrend’s Theorem. There exists a set $A \subset[1, N]$ with $|A| \gg B \exp (-c \sqrt{\log N})$ containing no non-trivial three term arithmetic progressions. In other words $r_3(N) \gg$ $N \exp (-c \sqrt{\log N})$

Roth’s proof is based on Fourier analysis. It falls naturally into two parts: either the set A looks random in which case we may easily count the number of three term progressions, or the set has some structure which can be exploited to find a subset with increased density. The crucial point is that the idea of randomness here can be made precise in terms of the size of the Fourier coefficients of the set. This argument is quite hard to generalize to four term progressions (or longer), and was only extended recently with the spectacular work of Gowers.

Returning to the Erdős-Turán conjecture, the next big breakthrough was made by Szemerédi who in 1969 established the case $k=4$, and in 1975 dealt with the general case $k \geq 5$. His proof was a tour-de-force of extremely ingenious and difficult combinatorics. One of his ingredients was van der Waerden’s theorem, and so this did not lead to a good bound there.

Szemerédi’s Theorem. Given $k$ and $\delta>0$, there exists $N=N(k, \delta)$ such that any set $A \subset[1, N]$ with $|A| \geq \delta N$ contains a non-trivial $k$ term arithmetic progression.

An entirely different approach was opened by the work of Furstenberg (1977) who used ergodic theoretic methods to obtain a new proof of Szemerédi’s theorem. The ergodic theoretic approach also did not lead to any good bounds, but was useful in proving other results previously inaccessible. For example, it led to a multi-dimensional version of Szemerédi’s theorem, also a density version of the Hales-Jewett theorem (due to Katznelson and Ornstein), and also allowed for the common difference of the APs to have special shapes (e.g. squares).

In 1998-2001 Gowers made a major breakthrough by extending Roth’s harmonic analysis techniques to prove Szemerédi’s theorem. This approach finally gave good bounds for the van der Waerden numbers.

# 加性组合代写

van der Waerden 的证明是通过巧妙的初等归纳论证得出的 $k$ 和 $r$. 证明并没有很好地说明有多大 $N(k, r)$ 需要是。Hales 和 Jewett (1963) 随后发现了一个更一般的结果，对 Shelah (1988) 进行了很好的改进，但 van der Waerden 数的界限同样很差。
Hales-Jewett 定理。让 $k$ 和 $r$ 被给予。存在一个数 $N=N(k, r)$ 这样如果点在 $[1, k]^N$ 使用着色 $r$ 颜色则有 一条单色的”组合线”。这里的组合线是 $k$ 以下类型的点: 某些坐标是固定的，并且某些非空坐标集被指定为 “通配符”，取值从 1 到 $k$.

Erdôs 和 Turan 提出了一种更强形式的范德瓦尔登定理，部分原因是布望解决更强大的问题会导致范德瓦 尔登定理的更好版本。

$r_3(N) \ll N \sqrt{\log \log N / \log N}$. 什么是真正的本质 $r_3(N)$ ? 如果我们选择一个随机集合 $A$ 在 $[1, N]$ 我们可能期望它有大约 $|A|^3 / N$ 三学期 $\mathrm{AP}$ 。这表明 $r_3(N)$ 也许是大小 $N^{1 / 3}$. 然而，在 1946 年， Behrend 发现了一种巧妙的结构，效果要好得多。

Roth 的证明基于傅里叶分析。它自然分为两部分：要么集合 $\mathrm{A}$ 看起来是随机的，在这种情况下我们可以 很容易地计算出三项级数的数量，要么集合具有某种结构，可以利用它来找到密度增加的子集。关键是这 里的随机性概念可以根据集合的傅立叶系数的大小来精确化。这个论点很难推广到四项级数 (或更长)， 并且最近才随着 Gowers 的出色工作而得到扩展。

Szemerédi 的定理。鉴于 $k$ 和 $\delta>0$ ，那里存在 $N=N(k, \delta)$ 这样任何集合 $A \subset[1, N]$ 和 $|A| \geq \delta N$ 包 含一个不平凡的 $k$ 术语等差级数。

Furstenberg (1977) 的工作开创了一种完全不同的方法，他使用遍历理论方法获得 Szemerédi 定理的新 证明。遍历理论方法也没有得出任何好的界限，但有助于证明以前无法获得的其他结果。例如，它导致了 Szemerédi 定理的多维版本，也是 Hales-Jewett 定理的密度版本（由于 Katznelson 和 Ornstein)，并 且还允许 AP 的公差具有特殊形状 (例如正方形) ).
1998-2001 年，Gowers 通过扩展 Roth 的调和分析技术来证明 Szemerédi 定理，取得了重大突破。这种 方法最终为 van der Waerden 数提供了良好的界限。

## 有限元方法代写

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## 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。