Pablo ROTONDO successfully defended his PhD thesis on September 27th, 2018, at Université Paris-Diderot.
Title: Probabilistic studies in Number Theory and Word Combinatorics: instances of dynamical analysis
Directors:
- Brigitte VALLÉE, Université de Caen, France
- Valérie BERTHÉ, IRIF, Université Paris-Diderot, France
- Alfredo VIOLA, Universidad de la República, Uruguay
Evaluation committee:
- Bruno SALVY (rapporteur)
- Pierre ARNOUX (rapporteur)
- Cyril NICAUD (examiner)
- Franco ROBLEDO (examiner)
- Thomas STOLL (examiner)
Abstract.
Dynamical Analysis incorporates tools from dynamical systems, namely the Transfer Operator, into the framework of Analytic Combinatorics, permitting the analysis of numerous algorithms and objects naturally associated with an underlying dynamical system. This dissertation presents, in the integrated framework of Dynamical Analysis, the probabilistic analysis of seemingly distinct problems in a unified way: the probabilistic study of the recurrence function of Sturmian words, and the probabilistic study of the Continued Logarithm algorithm.
Sturmian words are a fundamental family of words in Word Combinatorics. They are in a precise sense the simplest infinite words that are not eventually periodic. Sturmian words have been well studied over the years, notably by Morse and Hedlund (1940) who demonstrated that they present a notable number theoretical characterization as discrete codings of lines with irrational slope, relating them naturally to dynamical systems, in particular the Euclidean dynamical system. These words have never been studied from a probabilistic perspective. Here, we quantify the recurrence properties of a “random” Sturmian word, which are dictated by the so-called “recurrence function”; we perform a complete asymptotic probabilistic study of this function, quantifying its mean and describing its distribution under two different probabilistic models, which present different virtues: one is a naturaly choice from an algorithmic point of view (but is innovative from the point of view of dynamical analysis), while the other allows a natural quantification of the worst-case growth of the recurrence function. We discuss the relation between these two distinct models and their respective techniques, explaining also how the two seemingly different techniques employed could be linked through the use of the Mellin transform. In this dissertation we also discuss our ongoing work regarding two special families of Sturmian words: those associated with a quadratic irrational slope, and those with a rational slope (not properly Sturmian). Our work seems to show the possibility of a unified study.
The Continued Logarithm Algorithm, introduced by Gosper in Hakmem (1978) as a mutation of classical continued fractions, computes the greatest common divisor of two natural numbers by performing division-like steps involving only binary shifts and substractions. Its worst-case performance was studied recently by Shallit (2016), who showed a precise upper-bound for the number of steps and gave a family of inputs attaining this bound. In this dissertation we employ dynamical analysis to study the average running time of the algorithm, giving precise mathematical constants for the asymptotics, as well as other parameters of interest. The underlying dynamical system is akin to the Euclidean one, and was first studied by Chan (around 2005) from an ergodic, but the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying this system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the algorithm by Dynamical Analysis.