ICC   25427
INSTITUTO DE INVESTIGACION EN CIENCIAS DE LA COMPUTACION
Unidad Ejecutora - UE
artículos
Título:
On Simply Normal Numbers to different bases
Autor/es:
SLAMAN, THEODORE A.; BUGEAUD, YANN; BECHER, VERĂ“NICA
Revista:
MATHEMATISCHE ANNALEN
Editorial:
SPRINGER
Referencias:
Lugar: Berlin; Año: 2016 vol. 364 p. 125 - 150
ISSN:
0025-5831
Resumen:
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect powers, hence X is the set {2,3, 5,6,7,10,11,...} . Let M be a function from X to sets of positive integers such that, for each s in X, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases s^m such that s is in X and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.