[25] A sequence over a b-ary alphabet is normal in base b if every block of length k occurs (possibly overlapping) with limiting frequency 2−k. A sequence is normal if b = 2. A sequence is absolutely normal if it is normal in every base. A sequence (in base 2) is block-normal if for every m the sequence of m-bit strings obtained by splitteng the original sequence in m-bit blocks contains every m-bit block with limit frequency 2−m.

(a) Show that a sequence is block-normal iff it is normal.

(b) Show that the same sequences are normal in base b and base bh for every h.

(c) Show that a sequence is normal (in any finite base) if it is Martin-L¨of random.

(d) Show that a computable absolutely normal sequence (equivalent number) exists.
Comments. Source: [A.K. Shen, V.A. Uspensky, and N.K. Vereshchagin, Kolmogorov Complexity and Algorithmic Randomness, American Mathematical Society, 2017]. Hint for Item (b): use Item (a). Hint for Item (c): this follows from the definition of randomness. Hint for Item (d):
sequences that are not normal in a base b form a Schnorr effective null set [C.P. Schnorr, Lect. Notes Math., Vol. 218, Springer-Verlag, 1971]
as in Exercise 2.5.21 on page 166. Therefore, the union of these sets is also a Schnorr effective null set. Hence there exists a computable point outside it. The set of sequences that are normal in base b depends on b.
As a consequence, there are sequences that are not absolutely normal.
This is a nontrivial number-theoretic result and therefore outside the scope of this book.