Two Arithmetical Sources and Their Associated Tries

Authors Valérie Berthé, Eda Cesaratto, Frédéric Paccaut, Pablo Rotondo, Martín D. Safe, Brigitte Vallée

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Author Details

Valérie Berthé
  • Université de Paris, CNRS, IRIF, F-75006, Paris, France
Eda Cesaratto
  • CONICET and Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina
Frédéric Paccaut
  • LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, Amiens, France
Pablo Rotondo
  • LITIS, Université de Rouen, France
Martín D. Safe
  • Departamento de Matemática, Universidad Nacional del Sur (UNS), Bahía Blanca, Argentina
  • INMABB, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, Argentina
Brigitte Vallée
  • GREYC, CNRS and Université de Caen, France

Cite AsGet BibTex

Valérie Berthé, Eda Cesaratto, Frédéric Paccaut, Pablo Rotondo, Martín D. Safe, and Brigitte Vallée. Two Arithmetical Sources and Their Associated Tries. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


This article is devoted to the study of two arithmetical sources associated with classical partitions, that are both defined through the mediant of two fractions. The Stern-Brocot source is associated with the sequence of all the mediants, while the Sturm source only keeps mediants whose denominator is "not too large". Even though these sources are both of zero Shannon entropy, with very similar Renyi entropies, their probabilistic features yet appear to be quite different. We then study how they influence the behaviour of tries built on words they emit, and we notably focus on the trie depth. The paper deals with Analytic Combinatorics methods, and Dirichlet generating functions, that are usually used and studied in the case of good sources with positive entropy. To the best of our knowledge, the present study is the first one where these powerful methods are applied to a zero-entropy context. In our context, the generating function associated with each source is explicit and related to classical functions in Number Theory, as the ζ function, the double ζ function or the transfer operator associated with the Gauss map. We obtain precise asymptotic estimates for the mean value of the trie depth that prove moreover to be quite different for each source. Then, these sources provide explicit and natural instances which lead to two unusual and different trie behaviours.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • Theory of computation → Randomness, geometry and discrete structures
  • Combinatorics of words
  • Information Theory
  • Probabilistic analysis
  • Analytic combinatorics
  • Dirichlet generating functions
  • Sources
  • Partitions
  • Trie structure
  • Continued fraction expansion
  • Farey map
  • Sturm words
  • Transfer operator


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