Recall the definition of a normal number: a number aa is called normal to base bb if in its expansion in base bb, the number of appearances of every single single string of kk base-bb digits in its first nn digits divided by nn tends to 1bk\frac1{b^k} as nn tends to infinity. A number is called absolutely normal or just normal if it is normal to every base b≥2b \geq 2.

Given a fixed base bb, there is a wealth of explicit examples of numbers normal to base bb, such as a base bb Champernowne or Copeland-Erdإ‘s constant. There are some more elusive examples of absolutely normal numbers: any Chaitin’s constant is normal, but such numbers are always uncomputable; apparently, a computable normal number is constructed in this article, although it does not actually calculate any of its digits.

Now, my question: are there numbers which are normal to one base, but not another? If so, can we explicitly give an example of such a number?

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1 Answer

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There are infinitely many such numbers. One family of such numbers are the Stoneham constants, defined as

αb,c=∑n≥11cnbcn

\alpha_{b,c} = \sum_{n \ge 1}\frac{1}{c^n b^{c^n}}

where gcd(b,c)=1\gcd(b,c) = 1. Stoneham constants are known to be normal in base bb but not in base B=bpcqrB = b^p c^q r where p,q,r≥1p,q,r \ge 1 and neither bb nor cc divide rr.

For an more details explicit construction of such numbers, you can refer to the paper by David H. Bailey and Jonathan M. Borwein.

http://www.davidhbailey.com/dhbpapers/nonnormality.pdf