arXiv subjects

Eda Cesaratto

Publications and source records attributed to Eda Cesaratto.

11 recordsLinked to original sources

arXiv2025

Gaps in binary cyclotomic polynomials

For odd prime numbers $p < q$, let $\Phi_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $\Phi_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder of $q$ divided by $p$. For $q$ congruent to $\pm 1$ modulo $p$, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of $\Phi_{pq}$ are described as concatenations of words arising from iterations of a circular map.

math.NT↗

arXiv2025

Shannon Weights for binary dynamical recurrent sources of zero entropy

A probabilistic source is defined as the set of infinite words (over a given denumerable alphabet) endowed with a probability $\mu$. The paper deals with general binary sources where the distribution of any symbol (0 or 1) may depend on an unbounded part of the previous history. The paper studies Shannon weights: whereas the classical Shannon entropy ${\cal E}_{\mu}$ is the average amount of information brought by one symbol of the emitted word, the Shannon weight sequence deals with the average amount of information $m_{\mu}(n)$ that is brought by the emitted prefix of length $n$. For a source with a non zero entropy, the estimate $m_{\mu}(n)\sim{\cal E}_{\mu} \cdot n$ thus holds. The paper considers the model of dynamical sources, where a source word isemitted as an encoded trajectory of a dynamical system of the unit interval, when endowed with probability $\mu$. It focus on sources with zero entropy and gives explicit constructions for sources whose Shannon weight sequence satisfies $m_{\mu}(n)=o(n)$, with a prescribed behaviour. In this case, sources with zero entropy lead to dynamical systems built on maps with an indifferent fixed point. This class notably contains the celebrated Farey source, which presents well-known intermittency phenomena. Methods are based on analytic combinatorics and generating functions, and they are enlarged, in the present dynamical case, with dynamical systems tools (mainly transfer operators).

math.DS↗

arXiv2024

Poisson genericity in numeration systems with exponentially mixing probabilities

We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotically follows a Poisson law as the block length increases. We prove that almost all sequences are Poisson generic. Our result generalizes Peres and Weiss' theorem about Poisson genericity of integral bases numeration systems. In particular, we obtain that their continued fraction expansions for almost all real numbers are Poisson generic.

math.PR↗

arXiv2022

Lochs-type theorems beyond positive entropy

Lochs' theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. In its original version, Lochs' theorem related decimal expansions with continued fraction expansions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. This is the viewpoint we develop here. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature.

math.NT↗

arXiv2021

Binary Cyclotomic Polynomials: Representation via Words and Algorithms

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.

math.NT↗

arXiv2021

On the number of words with restrictions on the number of symbols

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 Laplace's method for sums and known bounds of Stirling numbers of the second kind. We express our result in terms of inequalities.

math.CO↗

arXiv2017

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.

math.NT↗

arXiv2014

The distribution of factorization patterns on linear families of polynomials over a finite field

We obtain estimates on the number $|\mathcal{A}_{\boldsymbolλ}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern $\boldsymbolλ:=1^{λ_1}2^{λ_2}\cdots n^{λ_n}$. We show that $|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{n-m}+\mathcal{O}(q^{n-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbolλ)$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $\boldsymbolλ$ and $m$ is the codimension of $\mathcal{A}$. Furthermore, if the family $\mathcal{A}$ under consideration is "sparse", then $|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{n-m}+\mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $\mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$--notation in terms of $\boldsymbolλ$ and $\mathcal{A}$ with "good" behavior. Our approach reduces the question to estimate the number of $\mathbb{F}_q$--rational points of certain families of complete intersections defined over $\mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $\mathbb{F}_q$--rational points are established.

math.NT↗

arXiv2013

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})=μ_d.q+\mathcal{O}(1), where V(d,s,\bfs{a}) is such an average cardinality, μ_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.

math.NT↗

arXiv2013

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.

math.NT↗

arXiv2003

On the Hausdorff dimension of certain sets arising in Number Theory

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 $n_{_N}\le m \forall N\in \zN$, 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 ${\sum_{i=1}^N n_i\over N}\le m$.

math.NT↗