Is Π Normal?

A real number x is normal to base b if all finite words w over the alphabet {0,1, ..., b-1} occur as factors of the base-b expansion of x with a limiting frequency equal to b-|w|. A number is normal if it is normal in all integer bases b ≥ 2. Although there are some numbers, such as Chaitin's ω, which are known to be normal to all bases, and although it is known that almost all real numbers are normal to all bases, nothing is known about the "classical" numbers such as π, e, √2 and log 2.

Even much weaker questions, such as whether any particular factor occurs infinitely often in π, e, √2 and log 2, have no answers currently (except trivial ones, such as whether π contains infinitely many 1's in its base-2 expansion).

More generally, we could ask for the subword complexity of the base-b expansions of numbers in certain classes, such as the algebraic irrational numbers. Although we expect that that every subword eventually appears, and so the subword complexity would be bn, only much weaker results are known (see work of Adamczewski, Bugeaud, and co-authors) and these results require deep theorems of Diophantine approximation. For example, it is now known that the subword complexity p( n) for irrational algebraic numbers satisfies

limn → ∞ p( n)/n = +∞ .

-- JeffreyShallit - 13 Oct 2010

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Topic revision: r4 - 2018-02-22 - JeffreyShallit
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