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Random generators and normal numbers. (English) Zbl 1165.11328

Summary: Pursuant to the authors’ previous chaotic-dynamical model for random digits of fundamental constants [D. H. Bailey and R. E. Crandall, Exp. Math. 10, No. 2, 175–190 (2001; Zbl 1047.11073)], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results are achieved: We establish \(b\)-normality for constants of the form \(\sum_i 1/(b^{m_i} c^{n_i})\) for certain sequences \((m_i), (n_i)\) of integers. This work unifies and extends previously known classes of explicit normals. We prove that for coprime \(b,c>1\) the constant \(\alpha_{b,c} = \sum_{n=c,c^2,c^3,\dots} 1/(n b^n)\) is \(b\)-normal, thus generalizing the Stoneham class of normals [R. G. Stoneham, Acta Arith. 22, 277–286 (1973; Zbl 0276.10028)]. Our approach also reproves \(b\)-normality for the Korobov class [N. Korobov, Math. Notes 47, No. 2, 128–132 (1990); translation from Mat. Zametki 47, No. 2, 28–33 (1990; Zbl 0689.10059)] \(\beta_{b,c,d}\), for which the summation index \(n\) above runs instead over powers \(c^d, c^{d^2}, c^{d^3}, \dots\) with \(d>1\). Eventually we describe an uncountable class of explicit normals that succumb to the PRNG approach. Numbers of the \(\alpha, \beta\) classes share with fundamental constants such as \(\pi\), \(\log 2\) the property that isolated digits can be directly calculated, but for these new classes such computation tends to be surprisingly rapid. For example, we find that the googol-th (i.e., \(10^{100}\)-th) binary bit of \(\alpha_{2,3}\) is 0. We also present a collection of other results – such as digit-density results and irrationality proofs based on PRNG ideas – for various special numbers.

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11K45 Pseudo-random numbers; Monte Carlo methods
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