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A Formalisation of Gallagher’s Ergodic Theorem

Authors: Oliver Nash

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)


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.

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Oliver Nash. A Formalisation of Gallagher’s Ergodic Theorem. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 23:1-23:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{nash:LIPIcs.ITP.2023.23,
  author =	{Nash, Oliver},
  title =	{{A Formalisation of Gallagher’s Ergodic Theorem}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.23},
  URN =		{urn:nbn:de:0030-drops-183981},
  doi =		{10.4230/LIPIcs.ITP.2023.23},
  annote =	{Keywords: Lean proof assistant, measure theory, metric number theory, ergodicity, Gallagher’s theorem, Duffin-Schaeffer conjecture}
}
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