How much overlap?

0:00Covering space

Cover all of space with copies of one convex shape, moved around but never turned. Unless the shape tiles, the copies must overlap, so the cost is measured by density: the total volume of the copies per unit of space, the average number of copies over each point. Squares and hexagons tile, with density 1. Round disks cannot: Kershner showed the best density for disks is about 1.21, 2π/√27, and our lattice search agrees. The question here is about high dimensions. Give every convex body its best arrangement. How large can the density still be forced to be, for the worst body in n dimensions? Rogers showed in 1957 that every convex body in dimension 3 and up has a covering of density at most n log n + n log log n + 5n. Random centers show where n log n comes from. If copies of total density θ are dropped at random, a fraction e^(−θ) of space stays uncovered, and our simulation in 3 dimensions matches this. To fill the last holes by enlarging the body by a factor 1 + 1/n, which multiplies its volume by less than e, it is enough for the uncovered fraction to be below about n^(−n), and that takes density about n log n. Rogers’ centers may form several shifted copies of a lattice. A single lattice is harder: Rogers’ lattice bound grew like n^(log₂ log n), Ordentlich, Regev and Weiss brought it down to a constant times n², and earlier this year Li and Liu reached n log n times a power of log log n. They also found symmetric bodies whose lattice coverings need at least a constant times n log n. Two questions remained. Can the extra log log factor for single lattices be removed? And could a cleverer arrangement, with many lattice cosets or no pattern at all, do better than n log n for every body, perhaps even a constant times n? Naszódi recorded that linear-bound question. For Euclidean balls the gap is still wide: Coxeter, Few and Rogers proved a lower bound of order n, and Dumer an upper bound of about ½ n log n.

2:13The claims

Result 092 is two OpenAI preprints, both dated September 23. The first, 23 pages, claims that every convex body in n dimensions, for n ≥ 2, has a covering by translates along one lattice with density at most an absolute constant times n log n, with no symmetry or smoothness assumed. The second, 30 pages, claims that for every large n, some centrally symmetric convex body has translative covering density above c·n log n, for an absolute constant c, against every arrangement of centers, lattice or not. Together they pin down the worst covering density as order n log n, for lattice and unrestricted coverings, and for all bodies or only symmetric ones. The constants are not computed, and the result concerns the worst body in each dimension, not the covering density of any particular body. Euclidean balls are not settled. The bodies in the lower bound can even be taken to be symmetric polytopes with rational vertices. OpenAI also published Lean formalizations: the single-lattice theorem, about 26,000 lines, and the optimal order for all four worst-case densities, with the translative lower bound, about 14,000 more, matched against separate statements. We have not run the checks.

3:37How the proofs go

The lattice proof develops a framework of Li and Liu. Split the coordinates into a large horizontal part and a much smaller vertical part. After an affine change, a theorem of Eldan and Klartag says the horizontal slices of the body have volumes close to a Gaussian profile in the vertical direction. A mean-hole estimate of Rogers and Schmidt, in the form of Ordentlich, Regev and Weiss, gives one horizontal lattice that leaves only an exponentially small fraction uncovered in every slice on a finite list. Then one shear of a vertical lattice has to spread these slices so that every height gets enough horizontal coverage. Li and Liu’s binary corrections give each vertical coordinate two adjacent integer choices. The new paper keeps a whole hierarchy of these weighted branches, in blocks of sizes about b, b^0.9, b^0.81, and so on, down to a fixed cutoff. Each block loses only a tiny fraction of the Gaussian mass, so the losses stay bounded, while every single piece stays small enough for the mean-hole estimate. A random subset-sum sampler makes each block work from any starting point, and one shear handles all the patterns at once. The last holes go by a classical lemma going back to Rogers: if the uncovered fraction is below about k^(−n), enlarging the body by a factor 1 + 1/k covers everything. In the plane, disks of radius 0.68 on the square grid leave 0.3 percent uncovered; the lemma allows k = 18, and the enlarged radius, 0.718, does cover, since the grid needs 0.707. The lower bound uses a random body: a ball slightly larger than the unit ball, cut by symmetric slabs whose directions are random Poisson points on the sphere. With positive probability it keeps a substantial volume. Any periodic covering of low density, localized to a small window and rounded, becomes one of a finite list of center patterns, fixed before the body is drawn. For each pattern, a hole appears at a target point if the random slabs cut that point out of every nearby copy. Using about e^(c n log n) well-separated targets is what lets the bound reach order n log n. Following Li and Liu, these conditions are grouped into blocks, and Janson’s inequality shows that each pattern covers the window with doubly exponentially small probability, small enough to beat the number of patterns. A periodization principle carries the result from periodic to all coverings.

6:21Where it stands

Neither preprint is peer reviewed, and neither has been revised. OpenAI’s Lean formalizations cover the single-lattice theorem and the optimal order; we have not run them. Both papers build directly on Li and Liu’s 2026 work, a preprint and a companion manuscript, which supply the horizontal and vertical framework, the binary corrections, the random slab bodies and the witness method. Neither paper is among those OSVerify reviewed, and we have seen no expert review. If they hold, the worst covering density in n dimensions grows like n log n, up to constant factors, whether the centers form one lattice or not. The constants, and the case of Euclidean balls, are not settled.