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 -distribution normal but not simply -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.
@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},
pages = {199--217},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {27},
number = {1},
year = {2015},
doi = {10.5802/jtnb.899},
mrnumber = {3346970},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.899/}
}
TY - JOUR AU - Bill Mance TI - On the Hausdorff dimension of countable intersections of certain sets of normal numbers JO - Journal de théorie des nombres de Bordeaux PY - 2015 SP - 199 EP - 217 VL - 27 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.899/ DO - 10.5802/jtnb.899 LA - en ID - JTNB_2015__27_1_199_0 ER -
%0 Journal Article %A Bill Mance %T On the Hausdorff dimension of countable intersections of certain sets of normal numbers %J Journal de théorie des nombres de Bordeaux %D 2015 %P 199-217 %V 27 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.899/ %R 10.5802/jtnb.899 %G en %F 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/articles/10.5802/jtnb.899/
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