{"id":"openai-math/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line","pdf":"https://github.com/openai/math/blob/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026","tags":["fractal-geometry","self-similar-measures","dynamical-systems","entropy"],"score":0,"title":"The entropy-rate dimension formula for self-similar measures on the line","domain":"Dynamical systems and ergodic theory","family":"148","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://x.com/ConstantinKogl/status/2107897017372492221","date":"2026-10-07","kind":"x","added":"2026-10-24","author":"Constantin Kogler","handle":"ConstantinKogl"},{"url":"https://arxiv.org/abs/2610.10511","date":"2026-10-07","kind":"web","site":"arXiv.org","text":"A generalization of the exact overlaps conjecture predicts a formula for the dimension of self-similar measures in terms of the random walk entropy and the Lyapunov exponent. We prove the exact overlaps conjecture and its generalized version in dimension one. The proof relies on a new entropy…","added":"2026-10-24","title":"The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line","author":"Samuel Kittle, Constantin Kogler","record":"arxiv/2610.10511"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=I%20was%20stunned%20by%20the%20claimed%20solution%20to%20determining%20the%20dimension%20of%20self-similar%20measures%20on%20the%20line","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"I was stunned by the claimed solution to determining the dimension of self-similar measures on the line (Item 148 in the Review) and related problems. I am heartened that the proof appears to stand on the shoulders of earlier mathematicians and live within the framework they developed. I will…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":true,"family_title":"The entropy-rate dimension formula for self-similar measures"}