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Multi-coverings by lattice translates of convex bodies

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometrymath.CO — Combinatoricsmath.NT — Number Theory

Contributed by Scott Kominers ↗

Version 1 / Oct 09, 2026 / CC BY-SA 4.0

Abstract

We show that for every convex body K⊂RnK\subset\mathbb R^n (n≥2n\ge2), every 0<ε<10<\varepsilon<1, and every prescribed density ρ≥Aεnlog⁡n\rho\ge A_\varepsilon n\log n, there is an arrangement of translates of KK along a single lattice with mean multiplicity exactly ρ\rho and multiplicity at least (1−ε)ρ(1-\varepsilon)\rho everywhere. Consequently, lattice kk-fold coverings exist with density at most max⁡{(1+ε)k,Cεnlog⁡n}\max\{(1+\varepsilon)k,C_\varepsilon n\log n\}, where CεC_\varepsilon depends only on ε\varepsilon. In sufficiently large dimension nn, a version with prescribed common covolume holds simultaneously for up to exp⁡(nc0)\exp(n^{c_0}) bodies whose uniform distributions have a common covariance matrix, with c0>0c_0>0 absolute. The proof builds on OpenAI's horizontal–vertical construction for single-lattice coverings and uses Gaussian layer averaging and Laplace bounds for covering counts.

Provenance statement

The analysis in this article was heavily LLM-assisted, building off of the construction in OpenAI's "single-lattice covering bound" preprint iteratively with my guidance, mostly using GPT-6 Astra (accessed in part via Poe with the support of Quora, where I am an advisor).

Tools used

OpenAI
GPTVersion 6 Astra
Anthropic
Claude OpusVersion 5.5
Quora
Poe

References

  1. M. J. Cohn, Multiple lattice covering of space , Proc. London Math. Soc. (3) 32 (1976), no. 1, 117--132, doi:10.1112/plms/s3-32.1.117 (https://doi.org/10.1112/plms/s3-32.1.117).DOI
  2. R. Eldan and B. Klartag, Pointwise estimates for marginals of convex bodies , J. Funct. Anal. 254 (2008), no. 8, 2275--2293, doi:10.1016/j.jfa.2007.08.014 (https://doi.org/10.1016/j.jfa.2007.08.014).DOI
  3. N. Frankl, J. Nagy, and M. Naszódi, Coverings: variations on a result of R ogers and on the E psilon-net theorem of H aussler and W elzl , Discrete Math. 341 (2018), no. 3, 863--874, doi:10.1016/j.disc.2017.11.017 (https://doi.org/10.1016/j.disc.2017.11.017).DOI
  4. H. Groemer, Multiple packings and coverings , Studia Sci. Math. Hungar. 21 (1986), no. 1--2, 189--200, Journal archive (https://real-j.mtak.hu/5463/1/StudScientMath_21.pdf).link
  5. H. Li and X. Liu, Nearly sharp bounds for lattice coverings by convex bodies , preprint, version 2, 2026, arXiv:2607.28429v2 (https://arxiv.org/abs/2607.28429v2).arXiv
  6. B. Lund, Two-sided linear hashing and quadratic density bounds for smooth lattice coverings , preprint, 2026, arXiv:2609.20351 (https://arxiv.org/abs/2609.20351).arXiv
  7. M. Naszódi and A. Polyanskii, Approximating set multi-covers , European J. Combin. 67 (2018), 174--180, doi:10.1016/j.ejc.2017.08.001 (https://doi.org/10.1016/j.ejc.2017.08.001).DOI
  8. OpenAI , A single-lattice covering bound of order nlog⁡nn\log n , OpenAI Math Release preprint, catalog family 092 (https://github.com/openai/math/blob/main/CONTENTS.md), 2026, version dated September 23 (https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf), released on October 6; all source references use this version, identified by PDF SHA-256 c7dc74641f89bce3f456410280429a47b3e214365a9467b494e14ee6b2348cbb .link
  9. — , Translative covering densities of order nlog⁡nn\log n , OpenAI Math Release preprint, catalog family 092 (https://github.com/openai/math/blob/main/CONTENTS.md), 2026, version dated September 23 (https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf), released on October 6.link
  10. O. Ordentlich, O. Regev, and B. Weiss, New bounds on the density of lattice coverings , J. Amer. Math. Soc. 35 (2022), no. 1, 295--308, doi:10.1090/jams/984 (https://doi.org/10.1090/jams/984).DOI
Show all 15 referencesShow fewer references
  1. — , Bounds on the density of smooth lattice coverings , J. Anal. Math. 156 (2025), 301--326, doi:10.1007/s11854-025-0367-2 (https://doi.org/10.1007/s11854-025-0367-2).DOI
  2. C. A. Rogers, A note on coverings , Mathematika 4 (1957), no. 1, 1--6, doi:10.1112/S0025579300001030 (https://doi.org/10.1112/S0025579300001030).DOI
  3. — , Lattice coverings of space: The M inkowski-- H lawka theorem , Proc. London Math. Soc. (3) 8 (1958), no. 3, 447--465, doi:10.1112/plms/s3-8.3.447 (https://doi.org/10.1112/plms/s3-8.3.447).DOI
  4. — , Lattice coverings of space , Mathematika 6 (1959), no. 1, 33--39, doi:10.1112/S002557930000190X (https://doi.org/10.1112/S002557930000190X).DOI
  5. W. M. Schmidt, The measure of the set of admissible lattices , Proc. Amer. Math. Soc. 9 (1958), no. 3, 390--403, doi:10.1090/S0002-9939-1958-0096638-X (https://doi.org/10.1090/S0002-9939-1958-0096638-X).DOI

Version history

  1. v1Submitted by Scott KominersInitial depositCurrentOct 08, 2026