## Note on normal numbers.(English)Zbl 0063.00962

From the text: D. G. Champernowne [J. Lond. Math. Soc. 8, 254–260 (1933; Zbl 0007.33701, JFM 59.0214.01)] proved that the infinite decimal $$0.123456789101112\cdots$$ was normal (in the sense of Borel) with respect to the base 10, a normal number being one whose digits exhibit a complete randomness.
More precisely a number is normal provided each of the digits $$0, 1, 2, \cdots, 9$$ occurs with a limiting relative frequency of $$1/10$$ and each of the $$10^k$$ sequences of $$k$$ digits occurs with the frequency $$10^{-k}$$. Champernowne conjectured that if the sequence of all integers were replaced by the sequence of primes then the corresponding decimal $$0.12357111317\cdots$$ would be normal with respect to the base 10. We propose to show not only the truth of his conjecture but to obtain a somewhat more general result, namely:
Theorem. If $$a_1, a_2, \dots,$$ is an increasing sequence of integers such that for every $$\theta < 1$$ the number of $$a$$’s up to $$N$$ exceeds $$N^\theta$$ provided $$N$$ is sufficiently large, then the infinite decimal $$0.a_1a_2a_3\cdots$$ is normal with respect to the base $$\beta$$ in which these integers are expressed.
On the basis of this theorem the conjecture of Champernowne follows from the fact that the number of primes up to $$N$$ exceeds $$cN'/\log N$$ for any $$c < 1$$ provided $$N$$ is sufficiently large. The corresponding result holds for the sequence of integers which can be represented as the sum of two squares since every prime of the form $$4k+1$$ is also of the form $$x^2+2$$ and the number of these primes up to $$N$$ exceeds $$c'N/\log N$$ for sufficiently large $$N$$ when $$c'<1/2$$.
The above theorem is based on the concept of $$(\varepsilon, k)$$-normality of A. S. Besicovitch [Math. Z. 39, 146–156 (1934; Zbl 0009.20002, JFM 60.0937.01)].

### MSC:

 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

### Citations:

Zbl 0007.33701; JFM 59.0214.01; Zbl 0009.20002; JFM 60.0937.01
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