An inequality for Gaussians on lattices

Oded Regev, Noah Stephens-Davidowitz

Research output: Contribution to journalArticlepeer-review


We show that for any lattice L ⊆ Rn and vectors x, y ∈ Rn, ρ(L + x)2ρ(L + y)2≤ρ(L)2ρ(L+x+y)ρ(L+x-y), where ρ is the Gaussian mass function ρ(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 at tori, and a positive correlation inequality for Gaussian measures on lattices.

Original languageEnglish (US)
Pages (from-to)749-757
Number of pages9
JournalSIAM Journal on Discrete Mathematics
Issue number2
StatePublished - 2017


  • Gaussian measure
  • Heat kernel
  • Lattices
  • Theta function

ASJC Scopus subject areas

  • General Mathematics


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