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Invariant densities for random β -expansions

Karma Dajani, Martijn de Vries (2007)

Journal of the European Mathematical Society

Let β > 1 be a non-integer. We consider expansions of the form ∑ i = 1 ∞ d i / β i , where the digits ( d i ) i ≥ 1 are generated by means of a Borel map K β defined on { 0 , 1 } ℕ × [ 0 , ⌊ β ⌋ ( β − 1 ) ] . We show existence and uniqueness of a K β -invariant probability measure, absolutely continuous with respect to m p ⊗ λ , where m p is the Bernoulli measure on { 0 , 1 } ℕ with parameter p ( 0 < p < 1 ) and λ is the normalized Lebesgue measure on [ 0 , ⌊ β ⌋ ( β − 1 ) ] . Furthermore, this measure is of the form m p ⊗ μ β , p , where μ β , p is equivalent to λ . We prove that the measure of maximal entropy and m p ⊗ λ are mutually singular. In...

Irreducible Sobol' sequences in prime power bases

Henri Faure, Christiane Lemieux (2016)

Acta Arithmetica

Sobol' sequences are a popular family of low-discrepancy sequences, in spite of requiring primitive polynomials instead of irreducible ones in later constructions by Niederreiter and Tezuka. We introduce a generalization of Sobol' sequences that removes this shortcoming and that we believe has the potential of becoming useful for practical applications. Indeed, these sequences preserve two important properties of the original construction proposed by Sobol': their generating matrices are non-singular...

Irregularities of continuous distributions

Michael Drmota (1989)

Annales de l'institut Fourier

This paper deals with a continuous analogon to irregularities of point distributions. If a continuous fonction x : [ 0 , 1 ] → X where X is a compact body, is interpreted as a particle’s movement in time, then the discrepancy measures the difference between the particle’s stay in a proper subset and the volume of the subset. The essential part of this paper is to give lower bounds for the discrepancy in terms of the arc length of x ( t ) , 0 ≤ t ≤ 1 . Furthermore it is shown that these estimates are the best possible despite of...

Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow

Luca Marchese (2012)

Bulletin de la Société Mathématique de France

We study a diophantine property for translation surfaces, defined in terms of saddle connections and inspired by classical Khinchin condition. We prove that the same dichotomy holds as in Khinchin theorem, then we deduce a sharp estimate on how fast the typical Teichmüller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one.

Khintchine types of translated coordinate hyperplanes

Felipe A. Ramírez (2015)

Acta Arithmetica

There has been great interest in developing a theory of "Khintchine types" for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes...

Kloosterman sums in residue classes

Valentin Blomer, Djordje Milićević (2015)

Journal of the European Mathematical Society

We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any fixed primitive congruence class up to bounds towards the Ramanujan conjecture.

Currently displaying 421 – 440 of 1160