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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.
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