Maximal polarization for periodic configurations on the real line

Abstract

We prove that among all periodic configurations $\Gamma$ of points on the real line $\mathbb{R}$ the quantities \min_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \alpha (x - \gamma)^2} \quad \mbox{and} \quad \max_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \alpha (x - \gamma)^2} are maximized and minimized, respectively, if and only if the points are equispaced points whenever the number of points per period is sufficiently large (depending on $\alpha$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$. The first result is shown using Fourier series, the second follows from work of Cohn-Kumar on universal optimality