论文标题
自相关不平等的扩展，并应用于添加剂组合学
Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics
论文作者
论文摘要
在2019年的论文中，Barnard和Steinerberger表明，对于$ f \ in l^1（\ Mathbf {r}）$，以下自相关不平等：\ begin {equation*} \ min_ {0 \ leq t \ leq 1} \ int_ \ mathbf {r} f（x）f（x+t）\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ leq \ leq \ 0.411 || f || f || _ _ _ = {l^1}^2，equact y equage by contert {quonts {quonds $ $ 0. $ 0. $ 0.除了自身的有趣和重要的情况外，诸如此类的不平等还具有附加组合设备的应用，在这些问题中，某些问题（例如差异基础最小的问题）可以通过类似于上述积分的卷积不平等来封装。巴纳德（Barnard）和斯坦纳伯格（Steinerberger）认为，未来的研究可能会侧重于极端的功能的存在（本身与布拉斯帕姆·莱布（Brascamp-Lieb）类型的不平等有关）。 我们表明，要使$ f $在上述规定下是极端，我们必须有\ begin {qore*} \ max_ {x_1 \ in \ MathBf {r}} \ min_ {0 \ leq t \ leq 1} \ left [f（x_1-t）+f（x_1+t）\ right] \ left [f（x_2-t）+f（x_2+t）\ right]。 \ end {equation*}我们得出此结果的中心技术是$ f $的局部扰动，以增加自相关的值，同时留下$ || f || f || _ {l^1} $不变。这些扰动方法可以扩展以检查自相关的更一般的概念。令$ d，n \ in \ mathbb {z}^+$，$ f \ in l^1 $，$ a $ be a $ d \ times n $矩阵带有真实条目和列的$ a_i $，for $ 1 \ leq i \ leq i \ leq n $，$ c $是一个常数。对于一定类的矩阵$ a $，我们证明了$ f $的必要条件，以极大地自相关的不平等现象\ begin {equination*} \ min_ {\ Mathbf {t} \ in [0,1]^d} \ int _ {\ MathBf {r}} \ prod_ {i = 1}^n \ f（x+ \ \ \ \ \ \ \ \ m马理|| f || _ {l^1}^n。 \ end {equation*}
In a 2019 paper, Barnard and Steinerberger show that for $f\in L^1(\mathbf{R})$, the following autocorrelation inequality holds: \begin{equation*} \min_{0 \leq t \leq 1} \int_\mathbf{R} f(x) f(x+t)\ \mathrm{d}x \ \leq\ 0.411 ||f||_{L^1}^2, \end{equation*} where the constant $0.411$ cannot be replaced by $0.37$. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics where some problems, such as those of minimal difference basis, can be encapsulated by a convolution inequality similar to the above integral. Barnard and Steinerberger suggest that future research may focus on the existence of functions extremizing the above inequality (which is itself related to Brascamp-Lieb type inequalities). We show that for $f$ to be extremal under the above, we must have \begin{equation*} \max_{x_1 \in \mathbf{R} }\min_{0 \leq t \leq 1} \left[ f(x_1-t)+f(x_1+t) \right] \ \leq\ \min_{x_2 \in \mathbf{R} } \max_{0 \leq t \leq 1} \left[ f(x_2-t)+f(x_2+t) \right] . \end{equation*} Our central technique for deriving this result is local perturbation of $f$ to increase the value of the autocorrelation, while leaving $||f||_{L^1}$ unchanged. These perturbation methods can be extended to examine a more general notion of autocorrelation. Let $d,n \in \mathbb{Z}^+$, $f \in L^1$, $A$ be a $d \times n$ matrix with real entries and columns $a_i$ for $1 \leq i \leq n$, and $C$ be a constant. For a broad class of matrices $A$, we prove necessary conditions for $f$ to extremize autocorrelation inequalities of the form \begin{equation*} \min_{ \mathbf{t} \in [0,1]^d } \int_{\mathbf{R}} \prod_{i=1}^n\ f(x+ \mathbf{t} \cdot a_i)\ \mathrm{d}x\ \leq\ C ||f||_{L^1}^n. \end{equation*}
