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Let $t_1, \ldots, t_n$ be jointly independent boolean random variables. In Corollary $1.9$ we established a large deviation inequality for the polynomial $t_1+\cdots+t_n$. In many applications, it is also of interest to obtain large deviation inequalities for more general polynomials $P\left(t_1, \ldots, t_n\right)$ of the boolean variables $t_1, \ldots, t_n$. One particularly important case is that of a boolean polynomial
$$X:=\sum_{A \in \mathcal{A}} \prod_{j \in A} t_j,$$
where $\mathcal{A}$ is some collection of non-empty subsets of $[1, n]$. Observe that boolean polynomials are automatically positive and monotone increasing, and hence any two boolean polynomials are positively correlated via the FKG inequality (Theorem 1.19). More generally, if $X$ and $Y$ are boolean polynomials, then $f(X)$ and $f(Y)$ will be positively correlated whenever $f$ is a monotone increasing or decreasing function. In particular, we see that
$$\mathbf{E}\left(e^{-s(X+Y)}\right) \geq \mathbf{E}\left(e^{-s X}\right) \mathbf{E}\left(e^{-s Y}\right)$$
for any real number $s$. Using this fact, the exponential moment method, and some additional convexity arguments, Janson [190] derived a powerful bound for the lower tail probability $\mathbf{P}(X \leq \mathbf{E}(X)-T)$ :

Theorem $1.28$ (Janson’s inequality) Let $t_1, \ldots, t_n, \mathcal{A}, X$ be as above. Then for any $0 \leq T \leq \mathbf{E}(X)$ we have the lower tail estimate
$$\mathbf{P}(X \leq E(X)-T) \leq \exp \left(-\frac{T^2}{2 \Delta}\right)$$

where
$$\Delta=\sum_{A, B \in \mathcal{A}: A \cap B \neq \emptyset} \mathbf{E}\left(\prod_{j \in A \cup B} t_j\right) .$$
In particular, we have
$$\mathbf{P}(X=0) \leq \exp \left(-\frac{\mathbf{E}(X)^2}{2 \Delta}\right)$$

In previous sections, we often considered a polynomial $Y=Y\left(t_1, \ldots, t_n\right)$ of $n$ independent random variables $t_1, \ldots, t_n$, and wished to control the tail distribution of $Y$. For instance Chernoff’s inequality shows that the polynomial $t_1+\cdots+t_n$ is concentrated around its mean, while Janson’s inequality shows that the values of certain polynomials (especially those of low degree) could very rarely be significantly less than the mean.

In this section, we present some further results of this type, that assert that certain polynomials with small degrees are strongly concentrated. These results can be seen as generalizing Chernoff’s bound, and also provide (in certain cases) the missing half (upper tail bound) of Janson’s inequality.

To motivate the results, let us first give a classical result which works for any function $Y$ (not just a polynomial) provided that the Lipschitz constant of $Y$ is small.
Lemma $1.34$ (Lipschitz concentration inequality) Let $Y:{0,1}^n \rightarrow \mathbf{R}$ be a function such that $\left|Y(t)-Y\left(t^{\prime}\right)\right| \leq K$ whenever $t, t^{\prime} \in{0,1}^n$ differ in only one coordinate. Then if $t_1, \ldots, t_n$ are independent boolean variables, we have
$$\mathbf{P}\left(\left|Y\left(t_1, \ldots, t_n\right)-\mathbf{E}\left(Y\left(t_1, \ldots, t_n\right)\right)\right| \geq \lambda K \sqrt{n}\right) \leq 2 e^{-\lambda^2 / 2}$$
for all $\lambda>0$.

## 加性组合代写

$$X:=\sum_{A \in \mathcal{A}} \prod_{j \in A} t_j,$$

(定理 1.19) 。而一般地说，如果 $X$ 和 $Y$ 是布尔多项式，那 $(X)$ 和 $f(Y)$ 将呈正相关，无论何时 $f$ 是单调迫增或递减函数。特别 是，我们看到
$$\mathbf{E}\left(e^{-s(X+Y)}\right) \geq \mathbf{E}\left(e^{-s X}\right) \mathbf{E}\left(e^{-s Y}\right)$$

$$\mathbf{P}(X \leq E(X)-T) \leq \exp \left(-\frac{T^2}{2 \Delta}\right)$$

$$\Delta=\sum_{A, B \in \mathcal{A}: A \cap B \neq \emptyset} \mathbf{E}\left(\prod_{j \in A \cup B} t_j\right)$$

$$\mathbf{P}(X=0) \leq \exp \left(-\frac{\mathbf{E}(X)^2}{2 \Delta}\right)$$

$$\mathbf{P}\left(\left|Y\left(t_1, \ldots, t_n\right)-\mathbf{E}\left(Y\left(t_1, \ldots, t_n\right)\right)\right| \geq \lambda K \sqrt{n}\right) \leq 2 e^{-\lambda^2 / 2}$$

## 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。

Posted on Categories:Additive Combinatorics, 加性组合, 数学代写

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## 数学代写|加性组合代写Additive Combinatorics代考|The exponential moment method

Chebyshev’s inequality shows that if one has control of the second moment $\operatorname{Var}(X)=\mathbf{E}\left(|X-\mathbf{E}(X)|^2\right)$, then a random variable $X$ takes the value $\mathbf{E}(X)+$ $O\left(\lambda \operatorname{Var}(X)^{1 / 2}\right)$ with probability $1-O\left(\lambda^{-2}\right)$. If one uses higher moments, one can obtain better decay of the tail probability than $O\left(\lambda^{-2}\right)$. In particular, if one can control exponential moments ${ }^1$ such as $\mathbf{E}\left(e^{t X}\right)$ for some real parameter $t$, then one can obtain exponential decay in upper and lower tail probabilities, since Markov’s inequality yields
$$\mathbf{P}(X \geq \lambda)=\mathbf{P}\left(e^{t X} \geq e^{t \lambda}\right) \leq \frac{\mathbf{E}\left(e^{t X}\right)}{e^{t \lambda}}$$
for $t>0$ and $\lambda \in \mathbf{R}$, and similarly
$$\mathbf{P}(X \leq-\lambda)=\mathbf{P}\left(e^{-t X} \geq e^{t \lambda}\right) \leq \frac{\mathbf{E}\left(e^{-t X}\right)}{e^{t \lambda}}$$
for the same range of $t, \lambda$. The quantity $\mathbf{E}\left(e^{t X}\right)$ is known as an exponential moment of $X$, and the function $t \mapsto \mathbf{E}\left(e^{t X}\right)$ is known as the moment generating function, thanks to the Taylor expansion
$$\mathbf{E}\left(e^{t X}\right)=1+t \mathbf{E}(X)+\frac{t^2}{2 !} \mathbf{E}\left(X^2\right)+\frac{t^3}{3 !} \mathbf{E}\left(X^3\right)+\cdots$$

## 数学代写|加性组合代写Additive Combinatorics代考|Sidon’s problem on thin bases

We now apply Chernoff’s inequality to the study of thin bases in additive combinatorics.

Definition $1.11$ (Bases) Let $B \subset \mathbf{N}$ be an (infinite) set of natural numbers, and let $k \in \mathbf{Z}{+}$. We define the counting function $r{k, B}(n)$ for any $n \in \mathbf{N}$ as
$$r_{k, B}(n):=\left|\left{\left(b_1, \ldots, b_k\right) \in B^k: b_1+\cdots+b_k=n\right}\right| .$$
We say that $B$ is a basis of order $k$ if every sufficiently large positive integer can be represented as sum of $k$ (not necessarily distinct) elements of $B$, or equivalently if $r_{k, B}(n) \geq 1$ for all sufficiently large $n$. Alternatively, $B$ is a basis of order $k$ if and only if $\mathbf{N} \backslash k B$ is finite.

Examples $1.12$ The squares $\mathbf{N}^{\wedge} 2={0,1,4,9, \ldots}$ are known to be a basis of order 4 (Legendre’s theorem), while the primes $P={2,3,5,7, \ldots}$ are conjectured to be a basis of order 3 (Goldbach’s conjecture) and are known to be a basis of order 4 (Vinogradov’s theorem). Furthermore, for any $k \geq 1$, the $k$ th powers $\mathbf{N}^{\wedge} k=\left{0^k, 1^k, 2^k, \ldots\right}$ are known to be a basis of order $C$ ( $k$ ) for some finite $C(k)$ (Waring’s conjecture, first proven by Hilbert). Indeed in this case, the powerful Hardy-Littlewood circle method yields the stronger result that $r_{m, \mathbf{N}^{\wedge} k}(n)=\Theta_{m, k}\left(n^{\frac{m}{k}-1}\right)$ for all large $n$, if $m$ is sufficiently large depending on $k$ (see for instance [379] for a discussion). On the other hand, the powers of $k$ $k^{\wedge} \mathbf{N}=\left{k^0, k^1, k^2, \ldots\right}$ and the infinite progression $k \cdot \mathbf{N}={0, k, 2 k, \ldots}$ are not bases of any order when $k>1$.

## 数学代写|加性组合代写Additive Combinatorics代考|The exponential moment method

$$\mathbf{P}(X \geq \lambda)=\mathbf{P}\left(e^{t X} \geq e^{t \lambda}\right) \leq \frac{\mathbf{E}\left(e^{t X}\right)}{e^{t \lambda}}$$

$$\mathbf{P}(X \leq-\lambda)=\mathbf{P}\left(e^{-t X} \geq e^{t \lambda}\right) \leq \frac{\mathbf{E}\left(e^{-t X}\right)}{e^{t \lambda}}$$

$$\mathbf{E}\left(e^{t X}\right)=1+t \mathbf{E}(X)+\frac{t^2}{2 !} \mathbf{E}\left(X^2\right)+\frac{t^3}{3 !} \mathbf{E}\left(X^3\right)+\cdots$$

## 数学代写|加性组合代写Additive Combinatorics代考|Sidon’s problem on thin bases

〈left 的分隔符缺失或无法识别

〈left 的分隔符缺失或无法识别 被认为是秩序的基础 $C(k)$ 对于一些有限的 $C(k)$ (Waring 猜想，首先由
Hilbert 证明) 。实际上，在这种情况下，强大的 Hardy-Littlewood 圆法产生的结果更强: $r_{m, \mathbf{N}}{ }^*(n)=\Theta_{m, k}\left(n \frac{m}{k}-1\right)$ 对于

left 的分隔符缺失或无法识别

## 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。

Posted on Categories:Additive Combinatorics, 加性组合, 数学代写

## avatest™帮您通过考试

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The following general notational conventions will be used throughout the book.
Sets and functions
For any set $A$, we use
$$A^d:=A \times \cdots \times A=\left{\left(a_1, \ldots, a_d\right): a_1, \ldots, a_d \in A\right}$$
to denote the Cartesian product of $d$ copies of $A$ : thus for instance $\mathbf{Z}^d$ is the $d$ dimensional integer lattice. We shall occasionally denote $A^d$ by $A^{\oplus d}$, in order to distinguish this Cartesian product from the $d$-fold product set $A^{\cdot d}=A \cdot \ldots \cdot A$ of $A$, or the $d$-fold powers $A^{\wedge} d:=\left{a^d: a \in A\right}$ of $A$.

If $A, B$ are sets, we use $A \backslash B:={a \in A: a \notin B}$ to denote the set-theoretic difference of $A$ and $B$; and $B^A$ to denote the space of functions $f: A \rightarrow B$ from $A$ to $B$. We also use $2^A:={B: B \subset A}$ to denote the power set of $A$. We use $|A|$ to denote the cardinality of $A$. (We shall also use $|x|$ to denote the magnitude of a real or complex number $x$, and $|v|=\sqrt{v_1^2+\cdots+v_d^2}$ to denote the magnitude of a vector $v=\left(v_1, \ldots, v_d\right)$ in a Euclidean space $\mathbf{R}^d$. The meaning of the absolute value signs should be clear from context in all cases.)

If $A \subset Z$, we use $1_A: Z \rightarrow{0,1}$ to denote the indicator function of $A$ : thus $1_A(x)=1$ when $x \in A$ and $1_A(x)=0$ otherwise. Similarly if $P$ is a property, we let $\mathbf{I}(P)$ denote the quantity 1 if $P$ holds and 0 otherwise; thus for instance $1_A(x)=\mathbf{I}(x \in A)$.

We use $\left(\begin{array}{l}n \ k\end{array}\right)=\frac{n !}{k !(n-k) !}$ to denote the number of $k$-element subsets of an $n$-element set. In particular we have the natural convention that $\left(\begin{array}{l}n \ k\end{array}\right)=0$ if $k>n$ or $k<0$.

We shall rely frequently on the integers $\mathbf{Z}$, the positive integers $\mathbf{Z}^{+}:={1,2, \ldots}$, the natural numbers $\mathbf{N}:=\mathbf{Z}_{\geq 0}={0,1, \ldots}$, the reals $\mathbf{R}$, the positive reals $\mathbf{R}^{+}:={x \in \mathbf{R}: x>0}$, the non-negative reals $\mathbf{R}_{\geq 0}:={x \in \mathbf{R}: x \geq 0}$, and the complex numbers $\mathbf{C}$, as well as the circle group $\mathbf{R} / \mathbf{Z}:={x+\mathbf{Z}: x \in \mathbf{R}}$.

For any natural number $N \in \mathbf{N}$, we use $\mathbf{Z}_N:=\mathbf{Z} / N \mathbf{Z}$ to denote the cyclic group of order $N$, and use $n \mapsto n \bmod N$ to denote the canonical projection from $\mathbf{Z}$ to $\mathbf{Z}_N$. If $q$ is a prime power, we use $F_q$ to denote the finite field of order $q$ (see Section 9.4). In particular if $p$ is a prime then $F_p$ is identifiable with $\mathbf{Z}_p$.

If $x$ is a real number, we use $\lfloor x\rfloor$ to denote the greatest integer less than or equal to $x$.

## 加性组合代写

\eft 的分隔符缺失或无法识别 $A^{-d}=A \cdot \ldots \cdot A$ 的 $A$ ，或者 $d-$ 倍数权力 $\backslash \operatorname{left}$ 的分隔符缺失或无法识别 $\quad$ 的 $A$.

$|v|=\sqrt{v_1^2+\cdots+v_d^2}$ 表示向量的大小 $v=\left(v_1, \ldots, v_d\right)$ 在欧几里得空间 $\mathbf{R}^d$. 在所有情兄下，绝对值符号的含义应从上下文中 清楚。)

## 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。