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On Square-Free Numbers

Adam Grabowski (2013)

Formalized Mathematics

In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of [19]. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges [1]) according to [18] which were absent in the Mizar Mathematical Library. Some of them were expressed in terms of clusters’ registrations, enabling automatization machinery available in the Mizar system. Our main result of the article...

On strong uniform distribution, II. The infinite-dimensional case

Y. Lacroix (1998)

Acta Arithmetica

We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or L ∞ , using the entropy method. It follows that such a chain with positive lower density is bad for L ∞ . There also exist such bad chains with zero density.

On subsequences of convergents to a quadratic irrational given by some numerical schemes

Benoît Rittaud (2010)

Journal de Théorie des Nombres de Bordeaux

Given a quadratic irrational α , we are interested in how some numerical schemes applied to a convenient function f provide subsequences of convergents to α . We investigate three numerical schemes: secant-like methods and formal generalizations, which lead to linear recurring subsequences; the false position method, which leads to arithmetical subsequences of convergents and gives some interesting series expansions; Newton’s method, for which we complete a result of Edward Burger [1] about the existence...

On substitution invariant Sturmian words: an application of Rauzy fractals

Valérie Berthé, Hiromi Ei, Shunji Ito, Hui Rao (2007)

RAIRO - Theoretical Informatics and Applications

Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give...

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