{"id":"openai-math/Translative-covering-densities-of-order-n-log-n","pdf":"https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026","tags":["convex-geometry","covering-density","discrete-geometry"],"score":0,"title":"Translative covering densities of order n log n","domain":"Convex and metric geometry","family":"092","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MG","summary":"For every sufficiently large dimension n , we construct a centrally symmetric convex body whose translative covering density exceeds $`c n\\log n`$, where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=OpenAI%20announces%20a%20universal%20lattice-covering%20bound","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"OpenAI announces a universal lattice-covering bound and matching examples even for unrestricted translative coverings , determining the worst-case order over convex bodies. Distinct distances and unit distances. These problems featured in János Pach’s guest post on Guth and Katz and in my earlier…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"}],"formalized":false,"family_title":"The optimal order of convex-body covering density"}