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Math-2504 · mathematics · Geometry

Optimal Covering Density of Euclidean Balls

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Determine the optimal asymptotic covering density of Euclidean space by congruent balls.

Formulation references: [S1], [S2].

Short English statement

Determine the optimal asymptotic covering density of Euclidean space by congruent balls.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 2

Math-2504-A1 · Literature report · Human + LLM · gpt-6 astra ·

Human contribution summary on Math-2504: Scott Duke Kominers

Human Contribution: Scott Duke Kominers

Multi-covering extension of OpenAI Result 092

Paper and attribution.
Scott Duke Kominers, Multi-coverings by lattice translates of convex bodies, hexagon:2610.00165v1, is a preliminary manuscript dated October 8, 2026. It develops the single-lattice construction in OpenAI's Result 092 into a bound for arbitrary covering multiplicity. The author credits substantial LLM assistance, principally GPT-6 Astra, while taking responsibility for the problem, methods, exposition, and any errors. This record is a summary of that authored contribution.
Lattice multi-coverings (Theorem 1.1).
For a convex body $K\subset\mathbb R^n$ and a full-rank lattice $\Lambda$, write \[ N_{\Lambda,K}(x)=\sum_{\lambda\in\Lambda}\mathbf 1_K(x-\lambda), \qquad \rho=\frac{\operatorname{vol}_n(K)}{\det\Lambda}. \] A lattice $k$-fold covering requires $N_{\Lambda,K}(x)\geq k$ at every $x$, with one translate for each distinct lattice point. Its density $\rho$ is the mean multiplicity. Let $\theta_{\mathrm{lat},k}(K)$ be the infimum of these densities. Theorem 1.1 states that, for every $\varepsilon>0$, a constant $C_\varepsilon$ depending only on $\varepsilon$ gives \[ \theta_{\mathrm{lat},k}(K) \leq \max\{(1+\varepsilon)k,\ C_\varepsilon n\log n\} \quad(n\geq2,\ k\geq1). \] No symmetry assumption is imposed on $K$. In particular, the density approaches the unavoidable lower bound $k$ uniformly over bodies and dimensions when $k/(n\log n)$ tends to infinity.
Prescribed density and simultaneous coverings (Theorem 1.2).
For one body and $0<\varepsilon<1$, every prescribed density $\rho\geq A_\varepsilon n\log n$ can be realized by a lattice satisfying $N_{\Lambda,K}(x)\geq(1-\varepsilon)\rho$ everywhere, in every dimension $n\geq2$. In sufficiently large dimension, the theorem also constructs one lattice of a prescribed covolume $\Delta$ for up to $\exp(n^{c_0})$ bodies, for an absolute $c_0>0$. This simultaneous statement assumes that their uniform distributions share a covariance matrix and that each mean $\rho_j=\operatorname{vol}_n(K_j)/\Delta$ exceeds the threshold. The bodies can have different volumes; their means need not agree. There is no matching pointwise upper bound on multiplicity or claim about a typical random lattice.
Worst-case order (Corollary 1.3).
Combining Theorem 1.1 with Li--Liu's earlier one-fold lattice lower bound, the paper obtains worst-case density of order $k+n\log n$, both over all convex bodies and over origin-symmetric convex bodies. This is a worst-case lattice statement; it does not identify the optimal one-fold translative density for a particular body such as a ball.
Outline of the argument.
The proof adapts OpenAI's horizontal--vertical decomposition of a lattice. Approximately Gaussian section volumes provide layer weights, and averaging a bounded terminal block over its cosets retains almost all of the mean while keeping individual selected weights small. Independent spatial thinning turns a mean-hole estimate into a Laplace-transform bound for section counts. A random shear combines those estimates to make the region of insufficient multiplicity very small. A small dilation eliminates that region; a final lattice rescaling gives the prescribed mean. Equal covariance permits a common normalization in the simultaneous version.
Scope relative to this problem.
The recorded problem asks for the optimal asymptotic density of a one-fold covering by congruent Euclidean balls. Specializing Theorem 1.1 to a ball gives a lattice $k$-fold covering with the stated density bound and, at $k=1$, an $O(n\log n)$ upper bound. For multiplicities large compared with $n\log n$, it gives density asymptotic to $k$. These conclusions are partial progress relative to the recorded one-fold question: they do not determine its optimal asymptotic density. Corollary 1.3 concerns worst-case convex bodies under the lattice restriction and supplies no matching one-fold lower bound for balls.
Status and provenance.
This site records the paper's reported results and has not independently reviewed the proof. The contribution is displayed in the Human Contribution collection, with its disclosed machine assistance; it is separate from the OpenAI claim and from any independent community review.
Sources.

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Math-2504-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2504

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.

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

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