{"paper":{"title":"An Inequality for Gaussians on Lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.NT"],"primary_cat":"math.PR","authors_text":"Noah Stephens-Davidowitz, Oded Regev","submitted_at":"2015-02-17T04:58:02Z","abstract_excerpt":"$ \\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 $\\vec{x}, \\vec{y} \\in \\R^n$, \\[ \\rho(\\lat + \\vec{x})^2 \\rho(\\lat + \\vec{y})^2 \\leq \\rho(\\lat)^2 \\rho(\\lat + \\vec{x} + \\vec{y}) \\rho(\\lat + \\vec{x} - \\vec{y}) \\; , \\] where $\\rho$ is the Gaussian measure $\\rho(A) := \\sum_{\\vec{w} \\in A} \\exp(-\\pi \\| \\vec{w} \\|^2)$. 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 t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04796","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}