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A dynamical interpretation of the global canonical height on an elliptic curve.

Experimental Mathematics

A fibered system associated with the prime number sequence. (Sur un système fibré lié à la suite des nombres premiers.)

Experimental Mathematics

A Gauss-Kuzmin theorem for the Rosen fractions

Journal de théorie des nombres de Bordeaux

Using the natural extensions for the Rosen maps, we give an infinite-order-chain representation of the sequence of the incomplete quotients of the Rosen fractions. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.

A Gauss-Kuzmin-Lévy theorem for a certain continued fraction.

International Journal of Mathematics and Mathematical Sciences

A new proof of a conjecture of Yoccoz

Annales de l’institut Fourier

We give a new proof of the following conjecture of Yoccoz:$\left(\exists C\in ℝ\right)\phantom{\rule{3.33333pt}{0ex}}\left(\forall \theta \in ℝ\setminus ℚ\right)\phantom{\rule{1em}{0ex}}log\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}\Delta \left({Q}_{\theta }\right)\le -Y\left(\theta \right)+C,$where ${Q}_{\theta }\left(z\right)={\mathrm{e}}^{2\pi i\theta }z+{z}^{2}$, $\Delta \left({Q}_{\theta }\right)$ is its Siegel disk if ${Q}_{\theta }$ is linearizable (or $\varnothing$ otherwise), $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}\Delta \left({Q}_{\theta }\right)$ is the conformal radius of the Siegel disk of ${Q}_{\theta }$ (or $0$ if there is none) and $Y\left(\theta \right)$ is Yoccoz’s Brjuno function.In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz’s initial methods.We then extend this result to some new families of polynomials such as ${z}^{d}+c$ with $d>2$. We also show that...

A quantitative ergodic theory proof of Szemerédi's theorem.

The Electronic Journal of Combinatorics [electronic only]

A spanning set for the space of super cusp forms.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

A survey of results on density modulo $1$ of double sequences containing algebraic numbers

Acta Mathematica Universitatis Ostraviensis

In this survey article we start from the famous Furstenberg theorem on non-lacunary semigroups of integers, and next we present its generalizations and some related results.

A Wirsing-type approach to some continued fraction expansion.

International Journal of Mathematics and Mathematical Sciences

Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients

Journal de théorie des nombres de Bordeaux

In this paper the Engel continued fraction (ECF) expansion of any $x\in \left(0,1\right)$ is introduced. Basic and ergodic properties of this expansion are studied. Also the relation between the ECF and F. Ryde’s monotonen, nicht-abnehmenden Kettenbruch (MNK) is studied.

Analysis of two step nilsequences

Annales de l’institut Fourier

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg’s proof of Szemerédi’s Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for two step nilsequences and give a classification scheme for them.

Approximation properties of β-expansions

Acta Arithmetica

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_{i}\right)}_{i=1}^{\infty }\in {0,1}^{ℕ}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_{i}{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_{i}\right)}_{i=1}^{n}\in {0,1}^{n}$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :ℕ\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: ${W}_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_{i}\right)}_{i=1}^{n}\in {0,1}^{n}}\left[{\sum }_{i=1}^{n}\left({ϵ}_{i}\right)/\left({\beta }^{i}\right),{\sum }_{i=1}^{n}\left({ϵ}_{i}\right)/\left({\beta }^{i}\right)+\Psi \left(n\right)\right]$In other words, ${W}_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^{n}\left({ϵ}_{i}\right)/\left({\beta }^{i}\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^{n}\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure of ${W}_{\beta }\left(\Psi \right)$ is...

Arithmetic progressions and the primes.

Collectanea Mathematica

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

Aspects of uniformity in recurrence

Colloquium Mathematicae

We analyze and cite applications of various, loosely related notions of uniformity inherent to the phenomenon of (multiple) recurrence in ergodic theory. An assortment of results are obtained, among them sharpenings of two theorems due to Bourgain. The first of these, which in the original guarantees existence of sets x,x+h,$x+{h}^{2}$ in subsets E of positive measure in the unit interval, with lower bounds on h depending only on m(E), is expanded to the case of arbitrary finite polynomial configurations...

Atomic surfaces, tilings and coincidences II. Reducible case

Annales de l’institut Fourier

The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that...

Acta Arithmetica

Integers

Acta Arithmetica

Binomial-Coefficient Multiples of Irrationals.

Monatshefte für Mathematik

Codages de rotations et phénomènes d'autosimilarité

Journal de théorie des nombres de Bordeaux

Nous étudions une classe de suites symboliques, les codages de rotations, intervenant dans des problèmes de répartition des suites ${\left(n\alpha \right)}_{n\in ℕ}$ et représentant une généralisation géométrique des suites sturmiennes. Nous montrons que ces suites peuvent être obtenues par itération de quatre substitutions définies sur un alphabet à trois lettres, puis en appliquant un morphisme de projection. L’ordre d’itération de ces applications est gouverné par un développement bi-dimensionnel de type “fraction continue”...

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