Barnard and Steinerberger [‘Three convolution inequalities on the real line with connections to additive combinatorics’, Preprint, 2019, arXiv:1903.08731] established the autocorrelation inequality $$\begin{eqnarray}\min _{0\leq t\leq 1}\int _{\mathbb{R}}f(x)f(x+t)\,dx\leq 0.411||f||_{L^{1}}^{2}\quad \text{for}~f\in L^{1}(\mathbf{R}),\end{eqnarray}$$ 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. We show that for $f$ to be extremal for this inequality, we must have $$\begin{eqnarray}\max _{x_{1}\in \mathbb{R}}\min _{0\leq t\leq 1}\left[f(x_{1}-t)+f(x_{1}+t)\right]\leq \min _{x_{2}\in \mathbb{R}}\max _{0\leq t\leq 1}\left[f(x_{2}-t)+f(x_{2}+t)\right].\end{eqnarray}$$ 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 extremise autocorrelation inequalities of the form $$\begin{eqnarray}\min _{\mathbf{t}\in [0,1]^{d}}\int _{\mathbb{R}}\mathop{\prod }_{i=1}^{n}~f(x+\mathbf{t}\cdot a_{i})\,dx\leq C||f||_{L^{1}}^{n}.\end{eqnarray}$$

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自相关不等式在加法组合学中的应用

Barnard 和 Steinerberger ['实线上的三个卷积不等式与加法组合学有关'，预印本，2019，arXiv:1903.08731] 建立了自相关不等式$$\begin{eqnarray}\min _{0\leq t\leq 1}\int _{\mathbb{R}}f(x)f(x+t)\,dx\leq 0.411||f|| _{L^{1}}^{2}\quad \text{for}~f\in L^{1}(\mathbf{R}),\end{eqnarray}$$其中常数$0.411$不能被替换$0.37$. 除了本身有趣和重要之外，诸如此类的不等式在加法组合学中也有应用。我们证明了$f$要使这种不等式达到极值，我们必须有$$\begin{eqnarray}\max _{x_{1}\in \mathbb{R}}\min _{0\leq t\leq 1}\left[f(x_{1}-t)+f( x_{1}+t)\right]\leq \min _{x_{2}\in \mathbb{R}}\max _{0\leq t\leq 1}\left[f(x_{2}- t)+f(x_{2}+t)\right].\end{eqnarray}$$我们得出这个结果的核心技术是局部扰动$f$增加自相关的值，同时离开$||f||_{L^{1}}$不变。这些扰动方法可以扩展到检查更一般的自相关概念。让$d,n\in \mathbb{Z}^{+}$,$f\in L^{1}$,$澳元做一个$d\次 n$具有真实条目和列的矩阵$a_{i}$为了$1\leq i\leq n$和$加元成为一个常数。对于一大类矩阵$澳元, 我们证明了必要条件$f$极端化形式的自相关不等式$$\begin{eqnarray}\min _{\mathbf{t}\in [0,1]^{d}}\int _{\mathbb{R}}\mathop{\prod }_{i=1} ^{n}~f(x+\mathbf{t}\cdot a_{i})\,dx\leq C||f||_{L^{1}}^{n}.\end{eqnarray}$ $