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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

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Version history

  1. v1Submitted by Scott KominersInitial depositCurrentOct 08, 2026