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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。