EXTENSIONS OF AUTOCORRELATION INEQUALITIES WITH APPLICATIONS TO ADDITIVE COMBINATORICS

Abstract

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

$$\begin{eqnarray}\min _{\mathbf{t}\in [0,1]^{d}}\int _{\mathbb{R}}\mathop{\prod }_{i=1}^{n}~f(x+\mathbf{t}\cdot a_{i})\,dx\leq C||f||_{L^{1}}^{n}.\end{eqnarray}$$