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Math-2504-A2 · revision r1 · on Math-2504

OpenAI claim summary for Math-2504

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The universal upper bound gives order $n\log n$ for balls as a special case. The matching worst-case lower bound uses constructed bodies rather than Euclidean balls, so the optimal asymptotic ball-covering density is not determined.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Optimal Covering Density of Euclidean Balls
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Optimal Covering Density of Euclidean Balls

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 every convex body in dimension at least two admits a single-lattice covering of density at most an absolute multiple of $n\log n$; In manuscript 2 (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.
An absolute constant $C>0$ is claimed such that for every dimension $n\ge2$ and every convex body $K\subset\mathbb R^n$ there is one full-rank lattice $\Lambda$ with \[ K+\Lambda=\mathbb R^n,\qquad |K|/\det\Lambda\le Cn\log n. \] The ratio is lattice covering density. Neither central symmetry nor smooth boundary is needed, and the covering uses one lattice rather than a union of independently translated lattices.
Manuscript 2.
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 hierarchy of branching lattice blocks supplies usable anchors. Folded Gaussian sampling and simultaneous shears cover a finite collection of horizontal patterns, after which a final small-hole removal produces an exact covering by one lattice.

For manuscript 2 (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: every convex body in dimension at least two admits a single-lattice covering of density at most an absolute multiple of $n\log n$.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 2 (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.

This coverage does not establish a formal proof of the complete problem as stated here.

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