Annales Mathématiques
Blaise Pascal
no. S2
Describability via ubiquity and eutaxy in Diophantine approximation
Arnaud Durand1
1 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay, France
Annales mathématiques Blaise Pascal, Tome 22 (2015) no. S2, pp. 1-149.

We present a comprehensive framework for the study of the size and large intersection properties of limsup sets that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of Lévy processes.

DOI : 10.5802/ambp.349
Classification : 11J82, 11J83, 28A78, 28A80, 60D05, 60G17, 60G51
Arnaud Durand 1

1 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay, France 
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Arnaud Durand. Describability via ubiquity and eutaxy in Diophantine approximation. Annales mathématiques Blaise Pascal, Tome 22 (2015) no. S2, pp. 1-149. doi : 10.5802/ambp.349. https://ambp.centre-mersenne.org/articles/10.5802/ambp.349/

[1] A. Baker; Wolfgang M. Schmidt Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc. (3), Volume 21 (1970), pp. 1-11 | MR | Zbl

[2] Julien Barral; Stéphane Seuret Heterogeneous ubiquitous systems in ℝ d and Hausdorff dimension, Bull. Braz. Math. Soc. (N.S.), Volume 38 (2007) no. 3, pp. 467-515 | DOI | MR | Zbl

[3] Julien Barral; Stéphane Seuret Ubiquity and large intersections properties under digit frequencies constraints, Math. Proc. Cambridge Philos. Soc., Volume 145 (2008) no. 3, pp. 527-548 | DOI | MR | Zbl

[4] Julien Barral; Stéphane Seuret A localized Jarník-Besicovitch theorem, Adv. Math., Volume 226 (2011) no. 4, pp. 3191-3215 | DOI | MR | Zbl

[5] Victor Beresnevich On approximation of real numbers by real algebraic numbers, Acta Arith., Volume 90 (1999) no. 2, pp. 97-112 | MR | Zbl

[6] Victor Beresnevich Application of the concept of regular systems of points in metric number theory, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk (2000) no. 1, p. 35-39, 140 | MR

[7] Victor Beresnevich; Detta Dickinson; Sanju Velani Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc., Volume 179 (2006) no. 846, pp. x+91 | DOI | MR | Zbl

[8] Victor Beresnevich; Sanju Velani A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2), Volume 164 (2006) no. 3, pp. 971-992 | DOI | MR | Zbl

[9] Jean Bertoin Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996, pp. x+265 | MR | Zbl

[10] A. S. Besicovitch Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers, J. London Math. Soc., Volume S1-9 (1934) no. 2, pp. 126 | DOI | MR | Zbl

[11] Hermine Biermé; Anne Estrade Covering the whole space with Poisson random balls, ALEA Lat. Am. J. Probab. Math. Stat., Volume 9 (2012), pp. 213-229 | MR | Zbl

[12] Y. Bugeaud; A. Durand Metric Diophantine approximation on the middle-third Cantor set (2015) (To appear in J. Eur. Math. Soc.)

[13] Yann Bugeaud Approximation by algebraic integers and Hausdorff dimension, J. London Math. Soc. (2), Volume 65 (2002) no. 3, pp. 547-559 | DOI | MR | Zbl

[14] Yann Bugeaud Approximation par des nombres algébriques de degré borné et dimension de Hausdorff, J. Number Theory, Volume 96 (2002) no. 1, pp. 174-200 | MR | Zbl

[15] Yann Bugeaud A note on inhomogeneous Diophantine approximation, Glasg. Math. J., Volume 45 (2003) no. 1, pp. 105-110 | DOI | MR | Zbl

[16] Yann Bugeaud Approximation by algebraic numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press, Cambridge, 2004, pp. xvi+274 | DOI | MR | Zbl

[17] Yann Bugeaud An inhomogeneous Jarník theorem, J. Anal. Math., Volume 92 (2004), pp. 327-349 | DOI | MR | Zbl

[18] Yann Bugeaud Intersective sets and Diophantine approximation, Michigan Math. J., Volume 52 (2004) no. 3, pp. 667-682 | DOI | MR | Zbl

[19] J. W. S. Cassels An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957, pp. x+166 | MR | Zbl

[20] Michael Drmota; Robert F. Tichy Sequences, discrepancies and applications, Lecture Notes in Mathematics, 1651, Springer-Verlag, Berlin, 1997, pp. xiv+503 | MR | Zbl

[21] Arnaud Durand Propriétés d’ubiquité en analyse multifractale et séries aléatoires d’ondelettes à coefficients corrélés, Université Paris 12 (France) (2007) (Ph. D. Thesis)

[22] Arnaud Durand Sets with large intersection and ubiquity, Math. Proc. Cambridge Philos. Soc., Volume 144 (2008) no. 1, pp. 119-144 | DOI | MR | Zbl

[23] Arnaud Durand Ubiquitous systems and metric number theory, Adv. Math., Volume 218 (2008) no. 2, pp. 368-394 | DOI | MR | Zbl

[24] Arnaud Durand Large intersection properties in Diophantine approximation and dynamical systems, J. Lond. Math. Soc. (2), Volume 79 (2009) no. 2, pp. 377-398 | DOI | MR | Zbl

[25] Arnaud Durand Singularity sets of Lévy processes, Probab. Theory Related Fields, Volume 143 (2009) no. 3-4, pp. 517-544 | DOI | MR | Zbl

[26] Arnaud Durand On randomly placed arcs on the circle, Recent developments in fractals and related fields (Appl. Numer. Harmon. Anal.), Birkhäuser Boston, Inc., Boston, MA, 2010, pp. 343-351 | DOI | MR | Zbl

[27] Arnaud Durand; Stéphane Jaffard Multifractal analysis of Lévy fields, Probab. Theory Related Fields, Volume 153 (2012) no. 1-2, pp. 45-96 | DOI | MR | Zbl

[28] Aryeh Dvoretzky On covering a circle by randomly placed arcs, Proc. Nat. Acad. Sci. U.S.A., Volume 42 (1956), pp. 199-203 | MR | Zbl

[29] P. Erdős Representations of real numbers as sums and products of Liouville numbers, Michigan Math. J., Volume 9 (1962), pp. 59-60 | MR | Zbl

[30] K. J. Falconer Classes of sets with large intersection, Mathematika, Volume 32 (1985) no. 2, p. 191-205 (1986) | DOI | MR | Zbl

[31] K. J. Falconer Sets with large intersection properties, J. London Math. Soc. (2), Volume 49 (1994) no. 2, pp. 267-280 | DOI | MR | Zbl

[32] Kenneth Falconer Fractal geometry, John Wiley & Sons, Inc., Hoboken, NJ, 2003, pp. xxviii+337 (Mathematical foundations and applications) | DOI | MR | Zbl

[33] Ai-Hua Fan; Jun Wu On the covering by small random intervals, Ann. Inst. H. Poincaré Probab. Statist., Volume 40 (2004) no. 1, pp. 125-131 | DOI | Numdam | MR | Zbl

[34] Stéphane Jaffard The multifractal nature of Lévy processes, Probab. Theory Related Fields, Volume 114 (1999) no. 2, pp. 207-227 | DOI | MR | Zbl

[35] Stéphane Jaffard On lacunary wavelet series, Ann. Appl. Probab., Volume 10 (2000) no. 1, pp. 313-329 | DOI | MR | Zbl

[36] Stéphane Jaffard Wavelet techniques in multifractal analysis, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2 (Proc. Sympos. Pure Math.), Volume 72, Amer. Math. Soc., Providence, RI, 2004, pp. 91-151 | MR | Zbl

[37] V. Jarník Diophantischen Approximationen und Hausdorffsches Mass, Mat. Sb., Volume 36 (1929), pp. 371-381

[38] V. Jarník Über die simultanen Diophantischen Approximationen, Math. Z., Volume 33 (1931) no. 1, pp. 505-543 | MR

[39] A. Khintchine Zur metrischen Theorie der diophantischen Approximationen, Math. Z., Volume 24 (1926) no. 1, pp. 706-714 | DOI | MR

[40] A. Khintchine Ein Satz über lineare diophantische Approximationen, Math. Ann., Volume 113 (1937) no. 1, pp. 398-415 | DOI | MR | Zbl

[41] J. F. C. Kingman Poisson processes, Oxford Studies in Probability, 3, The Clarendon Press, Oxford University Press, New York, 1993, pp. viii+104 (Oxford Science Publications) | MR | Zbl

[42] J. F. Koksma Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh. Math. Phys., Volume 48 (1939), pp. 176-189 | MR | Zbl

[43] J. Kurzweil On the metric theory of inhomogeneous diophantine approximations, Studia Math., Volume 15 (1955), pp. 84-112 | MR | Zbl

[44] J. Lesca Sur les approximations diophantiennes à une dimension, Université de Grenoble (1968) (Ph. D. Thesis)

[45] Kurt Mahler Zur Approximation der Exponentialfunktion und des Logarithmus., J. Reine Angew. Math., Volume 166 (1932), pp. 118-150 | DOI | MR | Zbl

[46] Benoit B. Mandelbrot Renewal sets and random cutouts, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 22 (1972), pp. 145-157 | MR | Zbl

[47] Pertti Mattila Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995, pp. xii+343 (Fractals and rectifiability) | DOI | MR | Zbl

[48] J. Neveu Processus ponctuels, École d’Été de Probabilités de Saint-Flour, VI—1976, Springer-Verlag, Berlin, 1977, p. 249-445. Lecture Notes in Math., Vol. 598 | MR | Zbl

[49] L. Olsen; Dave L. Renfro On the exact Hausdorff dimension of the set of Liouville numbers. II, Manuscripta Math., Volume 119 (2006) no. 2, pp. 217-224 | DOI | MR | Zbl

[50] Walter Philipp Some metrical theorems in number theory, Pacific J. Math., Volume 20 (1967), pp. 109-127 | MR | Zbl

[51] Marc Reversat Approximations diophantiennes et eutaxie, Acta Arith., Volume 31 (1976) no. 2, pp. 125-142 | MR | Zbl

[52] C. A. Rogers Hausdorff measures, Cambridge University Press, London-New York, 1970, pp. viii+179 | MR | Zbl

[53] Wolfgang M. Schmidt Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc., Volume 110 (1964), pp. 493-518 | MR | Zbl

[54] Wolfgang M. Schmidt Badly approximable systems of linear forms, J. Number Theory, Volume 1 (1969), pp. 139-154 | MR | Zbl

[55] L. A. Shepp Covering the circle with random arcs, Israel J. Math., Volume 11 (1972), pp. 328-345 | MR | Zbl

[56] L. A. Shepp Covering the line with random intervals, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 23 (1972), pp. 163-170 | MR | Zbl

[57] Vladimir G. Sprindžuk Metric theory of Diophantine approximations, V. H. Winston & Sons, Washington, D.C.; A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979, pp. xiii+156 | MR

[58] Claude Tricot Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc., Volume 91 (1982) no. 1, pp. 57-74 | DOI | MR | Zbl

[59] David Williams Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991, pp. xvi+251 | DOI | MR | Zbl

[60] Eduard Wirsing Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math., Volume 206 (1960), pp. 67-77 | MR | Zbl

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