On simply normal numbers to different bases

• Published in: Mathematische annalen Vol. 364; no. 1-2; pp. 125 - 150
• Main Authors: Becher, Verónica, Bugeaud, Yann, Slaman, Theodore A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2016; Springer Verlag
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-015-1209-9

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 S be the set of positive integers that are not perfect powers, hence S is the set { 2 , 3 , 5 , 6 , 7 , 10 , 11 , … } . Let M be a function from S to sets of positive integers such that, for each s in S , 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 S 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.