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5

Further explicit constructions of normal and non-normal numbers

In Section 4.2, we have constructed explicitly real numbers which are normal to a given base. In the first section of this chapter, we describe another class of explicit real numbers with the same property. Then, we discuss the existence of explicit examples of absolutely normal numbers.

DEFINITION 5.1. A real number is called absolutely normal if it is normal to every integer base b ≥ 2. A real number is called absolutely non-normal if it is normal to no integer base b ≥ 2.

We briefly and partially mention in Section 5.2 the point of view of complexity and calculability theory. Then, in Section 5.3, we give an explicit example of an absolutely non-normal irrational number. We ...

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