On the Hausdorff dimension of countable intersections of certain sets of normal numbers
Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 199-217.

On démontre que l’ensemble des nombres qui sont Q-normaux en distribution mais pas simplement Q-normaux en ratio est de dimension de Hausdorff maximale. Sous certaines conditions, on peut aussi démontrer que les intersections dénombrables de ces ensembles sont encore de dimension maximale, en dépit du fait qu’elles ne sont pas gagnantes (au sens de W. Schmidt). En conséquence, nous pouvons construire plusieurs exemples explicites de nombres qui sont simultanément normaux en distribution mais pas simplement normaux en ratio par rapport à certaines familles dénombrables de suites de base. De plus, on démontre que certains ensembles connexes sont soit gagnants, soit de première catégorie.

We show that the set of numbers that are Q-distribution normal but not simply Q-ratio normal has full Hausdorff dimension. It is further shown under some conditions that countable intersections of sets of this form still have full Hausdorff dimension even though they are not winning sets (in the sense of W. Schmidt). As a consequence of this, we construct many explicit examples of numbers that are simultaneously distribution normal but not simply ratio normal with respect to certain countable families of basic sequences. Additionally, we prove that some related sets are either winning sets or sets of the first category.

Reçu le : 2013-11-15
Accepté le : 2014-04-13
Publié le : 2015-05-21
DOI : https://doi.org/10.5802/jtnb.899
Classification : 11K16,  11A63
@article{JTNB_2015__27_1_199_0,
     author = {Bill Mance},
     title = {On the Hausdorff dimension of countable intersections of certain sets of normal numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {1},
     year = {2015},
     pages = {199-217},
     doi = {10.5802/jtnb.899},
     mrnumber = {3346970},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2015__27_1_199_0/}
}
Bill Mance. On the Hausdorff dimension of countable intersections of certain sets of normal numbers. Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 199-217. doi : 10.5802/jtnb.899. https://jtnb.centre-mersenne.org/item/JTNB_2015__27_1_199_0/

[1] C. Altomare and B. Mance, Cantor series constructions contrasting two notions of normality, Monatsh. Math 164, (2011), 1–22. | MR 2827169 | Zbl 1276.11128

[2] G. Cantor, Über die einfachen Zahlensysteme, Zeitschrift für Math. und Physik 14, (1869), 121–128. | JFM 02.0085.01

[3] P. Erdős and A. Rényi, On Cantor’s series with convergent 1/q n , Annales Universitatis L. Eötvös de Budapest, Sect. Math. (1959), 93–109. | Zbl 0095.26501

[4] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Inc., Hoboken, New Jersey, 2003. | MR 3236784 | Zbl 1285.28011

[5] J. Galambos, Representations of real numbers by infinite series, Lecture Notes in Math., vol. 502, Springer-Verlag, Berlin, Hiedelberg, New York, 1976. | MR 568141 | Zbl 0322.10002

[6] J. Hančl and R. Tijdeman, On the irrationality of Cantor series, J. reine angew Math. 571 (2004), 145–158. | MR 2070147 | Zbl 1049.11076

[7] N. Korobov, Concerning some questions of uniform distribution, Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), 215–238. | MR 37876 | Zbl 0036.31104

[8] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Dover, Mineola, NY, 2006.

[9] P. Laffer, Normal numbers with respect to Cantor series representation, Ph.D. thesis, Washington State University, Pullman, Washington, 1974.

[10] B. Mance, Number theoretic applications of a class of Cantor series fractal functions part I, Acta Mathematica Hungarica, 144, 2 (2014), 449–493. | MR 3274409

[11] —, Normal numbers with respect to the Cantor series expansion, Ph.D. thesis, The Ohio State University, Columbus, Ohio, 2010.

[12] —, Typicality of normal numbers with respect to the Cantor series expansion, New York J. Math. 17, (2011), 601–617. | MR 2836784

[13] —, Cantor series constructions of sets of normal numbers, Acta Arith. 156, (2012), 223–245. | MR 2999070 | Zbl 1276.11129

[14] N. G. Moshchevitin, On sublacunary sequences and winning sets (English), Math. Notes (2005), 3-4, 592–596. | MR 2226739 | Zbl 1151.11335

[15] H. Niederreiter, Almost-arithmetic progressions and uniform distribution, Trans. Amer. Math. Soc. 17 (1971), 283–292. | MR 284406 | Zbl 0219.10040

[16] P.E. O’Neil, A new criterion for uniform distribution, Proc. Amer. Math. Soc. 24, (1970), 1–5. | MR 248095 | Zbl 0224.10054

[17] A. Rényi, On the distribution of the digits in Cantor’s series, Mat. Lapok 7, (1956), 77–100. | MR 99968 | Zbl 0075.03703

[18] W. M. Schmidt, On normal numbers, Pacific J. Math. 10, (1960), 661–672. | MR 117212 | Zbl 0093.05401

[19] —, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123, (1966), 27–50. | MR 195595 | Zbl 0232.10029

[20] F. Schweiger, Über den Satz von Borel-Rényi in der Theorie der Cantorschen Reihen, Monatsh. Math. 74, (1969), 150–153. | MR 268150 | Zbl 0188.35103

[21] R. Tijdeman and P. Yuan, On the rationality of Cantor and Ahmes series, Indag. Math. 13 (3), (2002), 407–418. | MR 2057056 | Zbl 1018.11037

[22] T. Šalát, Über die Cantorschen Reihen, Czech. Math. J. 18 (93), (1968), 25–56. | MR 223305 | Zbl 0157.09904

[23] T. Šalát, Zu einigen Fragen der Gleichverteilung (mod 1), Czech. Math. J. 18 (93), (1968), 476–488. | MR 229586 | Zbl 0162.34701