Bounds on smoothness of lattice coverings

Paper Title:

Bounds on the density of smooth lattice coverings

Published on:

8 November 2023

Primary Category:

Number Theory

Paper Authors:

Or Ordentlich,

Oded Regev,

Barak Weiss

Bullets

Key Details

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Proves random lattices yield smooth covers for any convex shape in high dimensions

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Smoothness means covering density varies by at most epsilon across space

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Applies for shapes with volume at least polynomial in dimension n

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Builds on Dhar and Dvir's solution to the discrete Kakeya problem

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Quantifies how construction A lattices also give smooth covers

Explore the topics in this paper

convex geometry

high dimensions

lattice covering

randomness

uniform distribution

AI generated summary

Bounds on smoothness of lattice coverings

This paper proves that for any convex shape in high dimensions, a typical lattice can cover space with translates of that shape in a very uniform way. It shows that with high probability, a random lattice's covering density will not vary more than a small epsilon across all of space. The proofs rely on a breakthrough result of Dhar and Dvir connecting the Kakeya problem to randomness extractors.