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arXiv · 1311.0332

On Simply Normal Numbers to Different Bases

Verónica Becher·Yann Bugeaud·Theodore A. Slaman
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Abstract

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.

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Verónica Becher, Yann Bugeaud, Theodore A. Slaman. 2013-11-02. On Simply Normal Numbers to Different Bases. https://arxiv.org/abs/1311.0332

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