An inequality for Gaussians on lattices. Regev, O. & Stephens-Davidowitz, N. SIDMA, 2017.
An inequality for Gaussians on lattices [link]Paper  abstract   bibtex   44 downloads  
$ \newcommand{\R}{\ensuremath{ℝ}} \newcommand{łat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $łat ⊆ \R^n$ and vectors $x,y ∈ \R^n$, \[ f(łat + x)^2 f(łat + y)^2 ≤ f(łat)^2 f(łat + x + y) f(łat + x - y)   , \] where $f$ is the Gaussian measure $f(A) := ∑_{w ∈ A} \exp(-π \| w \|^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices.
@article{RSInequalityGaussians17,
	title = {An inequality for {Gaussians} on lattices},
	abstract = {$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $\lat \subseteq \R^n$ and vectors $x,y \in \R^n$, \[ f(\lat + x)^2 f(\lat + y)^2 \leq f(\lat)^2 f(\lat + x + y) f(\lat + x - y) \; , \] where $f$ is the Gaussian measure $f(A) := \sum_{w \in A} \exp(-\pi \| w \|^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices.},
	urldate = {2018-05-28},
	url = {http://arxiv.org/abs/1502.04796},
	journal = {SIDMA},
	author = {Regev, Oded and {Stephens-Davidowitz}, Noah},
	year = {2017},
		volume = {31},
	number = {2},
}

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