Gaussian Behavior of Quadratic Irrationals

We study the probabilistic behaviour of the continued fraction expansion of a quadratic irrational number, when weighted by some "additive" cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dynamical system associated with the Gauss map, and its weighted periodic trajectories. We work with analytic combinatorics methods, and mainly with bivariate Dirichlet generating functions; we use various tools, from number theory (the Landau Theorem), from probability (the Quasi-Powers Theorem), or from dynamical systems: our main object of study is the (weighted) transfer operator, that we relate with the generating functions of interest. The present paper exhibits a strong parallelism with the methods which have been previously introduced by Baladi and Vall\'ee in the study of rational trajectories. However, the present study is more involved and uses a deeper functional analysis framework.Comment: 39 pages In this second version, we have added an annex that provides a detailed study of the trace of the weighted transfer operator. We have also corrected an error that appeared in the computation of the norm of the operator when acting in the Banach space of analytic functions defined in the paper. Also, we give a simpler proof for Theorem

The number of reducible space curves over a finite field

"Most" hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible

On the value set of small families of polynomials over a finite field, I

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d,s,\bfs{a})=\mu_d.q+\mathcal{O}(1), where V(d,s,\bfs{a}) is such an average cardinality, \mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the \mathcal{O}--notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq--rational points is established.Comment: 30 page

Two Arithmetical Sources and Their Associated Tries

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 ? function, the double ? 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

Gaussian Behavior of Quadratic Irrationals

We study the probabilistic behaviour of the continued fraction expansion of a quadratic irrational number, when weighted by some "additive" cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dynamical system associated with the Gauss map, and its weighted periodic trajectories. We work with analytic combinatorics methods, and mainly with bivariate Dirichlet generating functions; we use various tools, from number theory (the Landau Theorem), from probability (the Quasi-Powers Theorem), or from dynamical systems: our main object of study is the (weighted) transfer operator, that we relate with the generating functions of interest. The present paper exhibits a strong parallelism with the methods which have been previously introduced by Baladi and Vallée in the study of rational trajectories. However, the present study is more involved and uses a deeper functional analysis framework.Fil: Cesaratto, Eda. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Vallée, Brigitte. Universite de Caen Basse Normandie; Franci

Gaussian behaviour of quadratic irrationals

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