Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector
of coefficients of the polynomial as a word on a ternary alphabet ${-1,0 ,+1}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.
We show that, in an alphabet of $n$ symbols, the number of words of length $n$ whose number of different symbols is away from $(1-1/e)n$, which is the value expected by the Poisson distribution, has exponential decay in $n$. We use Laplaces method fo
r sums and known bounds of Stirling numbers of the second kind. We express our result in terms of inequalities.
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 Vallee in the study of rational trajectories. However, the present study is more involved and uses a deeper functional analysis framework.
This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing $n_{_N}le m fo
rall Nin zN$, Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$, $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$, $F_m$ being the set of real numbers for which ${sum_{i=1}^N n_iover N}le m$.
