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Math-2648 · mathematics · Geometry

Worst-case symmetric translative covering density

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Are there constants $c>0$ and $n_{0}$ such that, for every $n\geq n_{0}$, some origin-symmetric convex body $K_n\subset R^{n}$ has translative covering density greater than c n log n? Translative covering density is the infimum of the asymptotic covering density over arbitrary coverings by translates of $K_n$, with multiplicities counted.

Short English statement

Are there constants $c>0$ and $n_{0}$ such that, for every $n\geq n_{0}$, some origin-symmetric convex body $K_n\subset R^{n}$ has translative covering density greater than c n log n? Translative covering density is the infimum of the asymptotic covering density over arbitrary coverings by translates of $K_n$, with multiplicities counted.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2648-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2648

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 092; 2026-09-23).

Worst-case symmetric translative covering density

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 092; 2026-09-23), the asserted result is that in every sufficiently large dimension, some symmetric convex body has translative covering density at least an absolute multiple of $n\log n$, ruling out a uniform linear bound.

Mathematical statement of the claimed result

Manuscript 1.
For a convex body $K\subset\mathbb R^n$, define its translative covering density by \[ \theta_T(K)=\inf_{\substack{X\subset\mathbb R^n\text{ locally finite}\\K+X=\mathbb R^n}} |K|\limsup_{R\to\infty}\frac{|X\cap[-R,R]^n|}{(2R)^n}. \] The covering centers are not required to form a lattice. The claimed lower bound is \[ \exists c>0\ \exists n_0\ \forall n\ge n_0\ \exists K_n=-K_n: \quad\theta_T(K_n)>c n\log n, \] with $c,n_0$ absolute and every $K_n$ a full-dimensional convex body.

Outline of the argument

For manuscript 1 (family 092; 2026-09-23), a random body is tested against a finite family obtained by rounding periodic covering patterns. Cap-intersection and Poisson-witness estimates make successful low-density coverage too unlikely for any pattern; periodization extends the obstruction to all coverings.

Formal verification

For manuscript 1 (family 092; 2026-09-23), OpenAI lists Lean statements for this assertion: in every sufficiently large dimension, some symmetric convex body has translative covering density at least an absolute multiple of $n\log n$, ruling out a uniform linear bound.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are CoveringDensity.lean, SingleLatticeCovering.lean.

Their named declarations include OAI.CoveringOrder.optimal_order, OAI.SingleLatticeCovering.single_lattice_covering.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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