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S-expansions in dimension two

Bernhard Schratzberger (2004)

Journal de Théorie des Nombres de Bordeaux

The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, can be used to transfer certain metrical and diophantine properties from one algorithm to the others. In...

Statistical convergence of subsequences of a given sequence

Martin Máčaj, Tibor Šalát (2001)

Mathematica Bohemica

This paper is closely related to the paper of Harry I. Miller: Measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811–1819 and contains a general investigation of statistical convergence of subsequences of an arbitrary sequence from the point of view of Lebesgue measure, Hausdorff dimensions and Baire’s categories.

Sur la mesure spectrale des suites multiplicatives

Jean Coquet (1979)

Annales de l'institut Fourier

Dans cet article, nous démontrons que la mesure spectrale d’une suite multiplicative de module ≤ 1 dont le spectre de Fourier-Bohr est non vide, est atomique. La preuve, basée sur un résultat de J.-P. Bertrandias, évite le calcul de la corrélation.

The complex sum of digits function and primes

Jörg M. Thuswaldner (2000)

Journal de théorie des nombres de Bordeaux

Canonical number systems in the ring of gaussian integers ℤ [ i ] are the natural generalization of ordinary q -adic number systems to ℤ [ i ] . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number b . In this paper we investigate the sum of digits function ν b of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the f -th power of a prime. Furthermore, we establish an Erdös-Kac type theorem...

The efficiency of approximating real numbers by Lüroth expansion

Chunyun Cao, Jun Wu, Zhenliang Zhang (2013)

Czechoslovak Mathematical Journal

For any x ∈ ( 0 , 1 ] , let x = 1 d 1 + 1 d 1 ( d 1 - 1 ) d 2 + ⋯ + 1 d 1 ( d 1 - 1 ) ⋯ d n - 1 ( d n - 1 - 1 ) d n + ⋯ be its Lüroth expansion. Denote by P n ( x ) / Q n ( x ) the partial sum of the first n terms in the above series and call it the n th convergent of x in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents { P n ( x ) / Q n ( x ) } n ≥ 1 in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of...

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