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A Proof of Shur’s Conjecture on the Growth of Power-Free Languages over Large Alphabets

Authors: Vuong Bui and Matthieu Rosenfeld

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)


Abstract
We settle a conjecture of Shur on an estimation of the exponential growth rates of the languages of (n/(n-1))-free words and (n/(n-1)) ^+-free words over large alphabets of size k with a correction of order O (1/(k²)).

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Vuong Bui and Matthieu Rosenfeld. A Proof of Shur’s Conjecture on the Growth of Power-Free Languages over Large Alphabets. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 32:1-32:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)


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@InProceedings{bui_et_al:LIPIcs.MFCS.2025.32,
  author =	{Bui, Vuong and Rosenfeld, Matthieu},
  title =	{{A Proof of Shur’s Conjecture on the Growth of Power-Free Languages over Large Alphabets}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{32:1--32:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.32},
  URN =		{urn:nbn:de:0030-drops-241398},
  doi =		{10.4230/LIPIcs.MFCS.2025.32},
  annote =	{Keywords: power-free languages, large alphabets, Shur’s conjecture, Dejean’s conjecture}
}
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