Describability via ubiquity and eutaxy in Diophantine approximation

Durand, Arnaud

Annales mathématiques Blaise Pascal, Numéro spécial Ecole de Printemps d’Analyse, 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.

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

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

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

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

[8] Beresnevich, Victor; Velani, Sanju 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] Bertoin, Jean Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996, pp. x+265 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[33] Fan, Ai-Hua; Wu, Jun 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] Jaffard, Stéphane The multifractal nature of Lévy processes, Probab. Theory Related Fields, Volume 114 (1999) no. 2, pp. 207-227 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

[47] Mattila, Pertti 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] Neveu, J. 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] Olsen, L.; Renfro, Dave L. 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] Philipp, Walter Some metrical theorems in number theory, Pacific J. Math., Volume 20 (1967), pp. 109-127 | MR | Zbl

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

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

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

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

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

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

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

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

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