Are there numbers normal to one base but not to another?

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?



1 Answer


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

\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.