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Maximal Polarization for Periodic Configurations on the Real Line

https://doi.org/10.1093/imrn/rnae003

Faulhuber, Markus; Steinerberger, Stefan (February 2024, International Mathematics Research Notices)

Abstract We prove that among all 1-periodic configurations $$\Gamma $$ of points on the real line $$\mathbb{R}$$ the quantities $$\min _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma )^{2}}$$ and $$\max _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma )^{2}}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $$n$$ per period is sufficiently large (depending on $$\alpha $$). This solves the polarization problem for periodic configurations with a Gaussian weight on $$\mathbb{R}$$ for large $$n$$. The first result is shown using Fourier series. The second result follows from the work of Cohn and Kumar on universal optimality and holds for all $$n$$ (independent of $$\alpha $$).

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