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Ubiquity and large intersections properties under digit frequencies constraints

Published online by Cambridge University Press:  01 November 2008

JULIEN BARRAL  and
STÉPHANE SEURET
JULIEN BARRAL
Affiliation:
Projet SISYPHE - INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. e-mail: [email protected]
STÉPHANE SEURET
Affiliation:
LAMA - CNRS UMR 8050 - Université Paris-Est - UFR Sciences et Technologie 61, avenue du Général de Gaulle, 94010 Créteil Cedex, France. e-mail: [email protected]
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Abstract

We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.

We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 527 - 548
Copyright
Copyright © Cambridge Philosophical Society 2008

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