The rabbit constant (related to Fibonacci numbers), as well as proof of its non-normalcy, is discussed in one of my recent articles, here. One might argue that it is a semi-artificial number. I was wondering if besides that number, and excluding artificial numbers such as $0.100111100000000111\ldots$, other irrational math constants are known to be non-normal (that is, with a digit distribution that is not uniform.)
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asked Apr 27, 2019 at 21:25
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1$\begingroup$ With $R = \sum_{n=1}^\infty 2^{-\lfloor n \phi \rfloor}$, $\phi=(1+\sqrt{5})/2$ then isn't it obvious it is non normal, since $\phi \in (1,2)$ means it doesn't have any $00$ in its binary expansion ? $\endgroup$– reunsApr 28, 2019 at 2:28
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3$\begingroup$ It is rather obvious, for a number of reasons, that it is not normal in base 2. What is less obvious is that these digits are also not normal in any base $b \in ]1, 2[$. In such as base, the distribution of the digits (if you pick up a number at random) is not uniform, and digits are negatively auto-correlated. The distribution of the digits and auto-correlations, in that case, depend on $b$, but not on the number itself if it is a "normal" number. The same is true for any number that is derived from the Beatty sequence using the same construct. $\endgroup$– Vincent GranvilleApr 28, 2019 at 4:39
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