{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-formalisation-of-gallagher-s-ergodic","title":"A formalisation of Gallagher's ergodic theorem","arxiv_id":"2302.00448","date":"2023-02-01","proceeding":null,"authors":["Oliver Nash"],"abstract":"Gallagher's ergodic theorem is a result in metric number theory. It states that the approximation of real numbers by rational numbers obeys a striking 'all or nothing' behaviour. We discuss a formalisation of this result in the Lean theorem prover. As well as being notable in its own right, the result is a key preliminary, required for Koukoulopoulos and Maynard's stunning recent proof of the Duffin-Schaeffer conjecture.","url_abs":"https://arxiv.org/abs/2302.00448v1","url_pdf":"https://arxiv.org/pdf/2302.00448v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-formalisation-of-gallagher-s-ergodic","repo_url":"https://github.com/leanprover-community/mathlib","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}