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 \(0, 1, \ldots , s-1\) occurs with the same frequency \(1/s\). Let \({\mathcal{S}}\) be the set of positive integers that are not perfect powers, hence \(\mathcal{S}\) is the set \(\{2,3, 5,6,7,10,11,\ldots \} \). Let \(M\) be a function from \(\mathcal{S}\) to sets of positive integers such that, for each \(s\) in \(\mathcal{S}\), 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 \(\mathcal{S}\) 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.
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Notes
To be accurate, the latter definition is not the one originally given by Borel, but equivalent to it.
Actually, Schmidt asserts the computability of \(c\) in separate paragraph (page 309 in the same article): “Wir stellen zunächst fest, daß man mit etwas mehr Mühe Konstanten \(a_{20}(r, s)\) aus Hilfssatz 5 explizit berechnen könnte, und daß dann \(x\) eine eindeutig definierte Zahl ist.”
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Becher, V., Bugeaud, Y. & Slaman, T.A. On simply normal numbers to different bases. Math. Ann. 364, 125–150 (2016). https://doi.org/10.1007/s00208-015-1209-9
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DOI: https://doi.org/10.1007/s00208-015-1209-9