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Discrepancy estimates for some linear generalized monomials

Roswitha Hofer, Olivier Ramaré (2016)

Acta Arithmetica

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, ( [ n α ] β ) n ≥ 0 , is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show...

Discrete planes, ℤ 2 -actions, Jacobi-Perron algorithm and substitutions

Pierre Arnoux, Valérie Berthé, Shunji Ito (2002)

Annales de l’institut Fourier

We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a ℤ 2 -action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of ℤ 2 -actions by rotations.

Distances dans la suite des multiples d'un point du tore à deux dimensions

Nicolas Chevallier (1996)

Acta Arithmetica

Introduction. Soit θ un élément de ¹=ℝ/ℤ. Considérons la suite des multiples de θ, x = ( n θ ) n ∈ ℕ . Pour tout n ∈ ℕ, ordonnons les n+1 premiers termes de cette suite, 0 = y₀ ≤ y₁ ≤...≤ yₙ ≤ 1 = pθ, p=0,...,n. La suite (y₀,...,yₙ) découpe l’intervalle [0,1] en n+1 intervalles qui ont au plus trois longueurs distinctes, la plus grande de ces longueurs étant la somme des deux autres. Cette propriété a été conjecturé par Steinhaus, elle est étroitement liée au développement en fraction continue de θ. On peut aussi...

Distribution des points de petite hauteur dans les groupes multiplicatifs

Francesco Amoroso, Sinnou David (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove a new lower bound for the height of points on a subvariety  V of a multiplicative torus, which lie outside the union of torsion subvarieties of  V . Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an ε the sharpest conjectures that can be formulated....

Dynamical directions in numeration

Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner (2006)

Annales de l’institut Fourier

This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β -numeration and its generalisations, abstract numeration systems and...

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