An Inequality for Gaussians on Lattices

@article{Regev2017AnIF,
  title={An Inequality for Gaussians on Lattices},
  author={Oded Regev and Noah Stephens-Davidowitz},
  journal={SIAM J. Discret. Math.},
  year={2017},
  volume={31},
  pages={749-757}
}
$ \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… 

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References

SHOWING 1-10 OF 27 REFERENCES

A Counterexample to Monotonicity of Relative Mass in Random Walks

For a finite undirected graph $G = (V,E)$, let $p_{u,v}(t)$ denote the probability that a continuous-time random walk starting at vertex $u$ is in $v$ at time $t$. In this note we give an example of

An inequality for the heat kernel on an Abelian Cayley graph

We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on

Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

A 2n+o(n)-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on n-dimensional Euclidean lattices and it is shown that the approximate closest vectors to a target vector t can be grouped into “lower-dimensional clusters,” and the discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP.

New bounds in some transference theorems in the geometry of numbers

The aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in R". To formulate the problem, we have to introduce some notation and terminology. We

A reverse Minkowski theorem

We prove a conjecture due to Dadush, showing that if ℒ⊂ ℝn is a lattice such that det(ℒ′) 1 for all sublattices ℒ′ ⊆ ℒ, then $$\sum_{y∈ℒ}^e-t2||y||2≤3/2,$$ where t := 10(logn + 2). From this we also

The Euclidean Distortion of Flat Tori

It is shown that for every n-dimensional lattice L the torus Rn/L can be embedded with distortion O(nċ√logn) into a Hilbert space and improves the exponential upper bound of O( n3n/2) and gets close to their lower bound of Ω(√n).

Fourier transforms with only real zeros

The class of even, nonnegative, finite measures p on the real line such that for any b > 0 the Fourier transform of exp(bt2) dp(t) has only real zeros is completely determined. This result is then

ON THE GAUSSIAN MEASURE OF THE INTERSECTION

The Gaussian correlation conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is

Worst-case to average-case reductions based on Gaussian measures

It is shown that solving modular linear equation on the average is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the rank of the lattice, and it is proved that the distribution that one obtains after adding Gaussian noise to the lattices has the following interesting property.

On the Closest Vector Problem with a Distance Guarantee

A substantially more efficient variant of the LLM algorithm is presented, and via an improved analysis, it is shown that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice.