= Absolutely normal number
= Absolute normality
{synonym}
An absolutely normal number is a <normal number> in every <integer> base $K\geq2$. This requires every finite <word> in every such <positional notation> to have its appropriate limiting <digit block frequency>. Knowing base-ten normality alone does not establish <absolute normality>; the base is part of the statement.
Borel's 1909 result says that <absolute normality> holds <almost everywhere> for <Lebesgue measure>: within $[0,1)$ the exceptions form a <Lebesgue null set>. In particular, the <set> of <absolutely normal numbers> has <Lebesgue measure> one there. This is a statement about the size of an exceptional <set>, not a test for a specified constant. https://doi.org/10.1007/BF03019651[Émile Borel, Les probabilités dénombrables et leurs applications arithmétiques, 1909]{ref} https://www.davidhbailey.com/dhbpapers/normal-pseudo.pdf[Bailey and Borwein, Normal Numbers and Pseudorandom Generators, 2011, introduction: simultaneous normality in all integer bases]{ref}
A modern route uses the <Birkhoff ergodic theorem> and <ergodicity of integer multiplication on the circle>. For each fixed base and digit <interval>, the time proportion equals its <Lebesgue measure> <almost everywhere>. The bases and finite <words> form <countable sets>, so a <countable intersection> of these measure-one <sets> still has <Lebesgue measure> one. For an <absolutely normal number>, the <word> $11$ therefore has frequency $1/4$ in base two, $1/9$ in base three, and $1/100$ in base ten. The same symbols represent different <intervals> in different bases.
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