{"id":"openai-math/A-power-saving-for-planar-unit-distances","pdf":"https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026","tags":["discrete-geometry","unit-distances","combinatorics"],"score":0,"title":"A power saving for planar unit distances","domain":"Combinatorics","family":"167","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove a power saving for the planar unit-distance problem: for some absolute $`\\beta\\lt 4/3`$, every set of n points in the Euclidean plane determines $`O(n^\\beta)`$ unordered pairs at unit distance.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"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=includes%20the%20sharp-order%20bound","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"includes the sharp-order bound for distinct distances in every fixed dimension and an improved planar unit-distance bound for some absolute . Erdős’s arithmetic-progression conjecture. My post on Bloom and Sisask’s breakthrough discussed the three-term case of the assertion that divergent…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans","author":"Gil Kalai"},{"url":"https://mathoverflow.net/a/515885","date":"2026-10-09","kind":"web","site":"mathoverflow.net","text":"According to the Oct 6 Open AI document dump, the upper bound can be lowered from $O(n^{4/3})$ to $O(n^{4/3-\\delta})$ for some positive (not explicitly computed) $\\delta$ . Source here ; I have not checked its accuracy.","added":"2026-11-04","title":"Answer to: What is the unit distance exponent?","author":"David E Speyer"},{"url":"https://arxiv.org/abs/2610.18089","date":"2026-10-13","kind":"web","site":"arXiv.org","text":"AI has become increasingly helpful in scientific discovery, yet human expert review remains essential for those problems without reliable automated evaluation, posing a fundamental challenge as candidate generation scales beyond available review capacity. This paper studies verification under…","added":"2026-10-24","title":"One-Sided Verification Under Limited Expert Review","author":"Fan Nie, Haotian Ye, Shurui Liu, Zihao Wang et al."}],"formalized":false,"family_title":"Planar distinct distances and unit-distance bounds"}