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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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References

REFERENCES

[1]Baker, A. and Schmidt, W. M.Diophantine approximation and Hausdorff dimension. Proc. London Math. Soc. 21 (1970), 1–11.Google Scholar
[2]Barral, J. and Seuret, S.Sums of Dirac masses and conditioned ubiquity. C. R. Acad. Sci. Paris, Ser. I 339 (2004), 787–792.CrossRefGoogle Scholar
[3]Barral, J. and Seuret, S.Heterogeneous ubiquitous systems and Hausdorff dimension in. Bull. Brazilian Math. Soc. 38 (3) (2007), 467–515.Google Scholar
[4]Beresnevich, V., Dickinson, H. and Velani, S.Measure theoretic laws for limsup sets. Mem. Amer. Math. Soc. 179, number 840 (2006), 1–91.Google Scholar
[5]Beresnevich, V. and Velani, S.A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. Math. 164 (2006), 971–992.CrossRefGoogle Scholar
[6]Bernik, V. I. and Tishchenko, K. I.Integral polynomials with an overfall of the coefficient values and Wirsing's theorem. Dokl. Akad. Nauk Belarusi 37 (1993), 9–11.Google Scholar
[7]Besicovitch, A. S.On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110 (1934), 321–330.CrossRefGoogle Scholar
[8]Brown, G., Michon, G. and Peyrière, J.On the multifractal analysis of measures. J. Stat. Phys. 66 (1992), 775–790.CrossRefGoogle Scholar
[9]Bugeaud, Y.An inhomogeneous Jarnik theorem. J. Anal. Math. 92 (2004), 327–349.Google Scholar
[10]Bugeaud, Y.Intersective sets and Diophantine approximation. Mich. Math. J. 52 (2004), 667–682.CrossRefGoogle Scholar
[11]Bugeaud, Y. and Teulié, O.Approximation d'un nombre réel par des nombres algébriques de degré donné. Acta Arith. 93 (2000), 77–86.CrossRefGoogle Scholar
[12]Bugeaud, Y.Diophantine approximation and Cantor sets. Math. Ann. 341 (2008), 677–684.CrossRefGoogle Scholar
[13]Cassels, J. W. S.An Introduction to Diophantine Approximation. (Cambridge University Press 1957).Google Scholar
[14]Dodson, M. M.Exceptional sets in dynamical systems and Diophantine approximation. Rigidity in Dynamics and Geometry (Cambridge, 2000), 77–98, (Springer, 2002).Google Scholar
[15]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika 37 (1990), 59–73.CrossRefGoogle Scholar
[16]Dodson, M. M., Melián, M. V., Pestana, D. and Vélani, S. L., Patterson measure and Ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 37–60.Google Scholar
[17]Durand, A.Ubiquitous systems and metric number theory. Adv. Math. 218 (2) (2008), 368–394.CrossRefGoogle Scholar
[18]Eggleston, H.The fractonial dimension of a set defined by decimal properties. Quart. J. Math. Oxford Ser. 20 (1949), 31–36.Google Scholar
[19]Falconer, K. J.Classes of sets with large intersection. Mathematika 32 (1985), 191–205.CrossRefGoogle Scholar
[20]Falconer, K. J.Sets with large intersection properties. J. London Math. Soc. (2)(49) (1994), 267–280.Google Scholar
[21]Falconer, K. J.Techniques in Fractal Geometry. (Wiley, 1997).Google Scholar
[22]Falconer, K. J.Representation of families of sets by measures, dimension spectra and Diophantine approximation. Math. Proc. Camb. Phil. Soc. 128 (2000), 111–121.Google Scholar
[23]Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers. (Oxford University Press, 1978).Google Scholar
[24]Hutchinson, J. E.Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[25]Jaffard, S.The multifractal nature of Lévy processes. Probab. Theory Relat. Fields 114 (1999), 207–227.CrossRefGoogle Scholar
[26]Jaffard, S.On lacunary wavelet series. Ann. Appl. Prob. 10 (2000), 313–329.CrossRefGoogle Scholar
[27]Jarnik, V.Diophantischen approximationen und Hausdorffsches mass. Mat. Sbornik 36 (1929), 371–381.Google Scholar
[28]Kingman, J. F.CCompletely random measures. Pacific J. Math. 21 (1967), 59–78.Google Scholar
[29]Levesley, J., Salp, C. and Velani, S.On a problem of K. Mahler: Diophantine approximation and Cantor sets. Math. Ann. 338 (1) (2007), 97–118Google Scholar
[30]Ma, J.-H., Wen, Z.-Y. and Wu, J.Besicovitch subsets of self-similar sets. Ann. Inst. Fourier, 52 (4) (2002), 1061–1074.CrossRefGoogle Scholar
[31]Mattila, P., Geometry of Sets and Measures in Euclidian Spaces. Cambridge Studies in Adv. Math. (Cambridge University Press 1995).CrossRefGoogle Scholar
[32]Shepp, L. A.Covering the line with random intervals. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 163–170.CrossRefGoogle Scholar