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Benford’s law for the \(3x+1\) function. (English) Zbl 1117.11018
The \(3x+1\) problem (sometimes called Collatz-problem) concerns the behaviour under iteration of the function \(T:\mathbb{N}\to\mathbb{N}\) defined by \(T(n)= \frac n2\) if \(n\) is even and \(T(n)=\frac{3n+1}{2}\) if \(n\) is odd. The \(3x+1\) conjecture asserts that when started from any \(n \in\mathbb{N}\) some \(k\) exists with iterate \(T^{(k)}(n)=1\) [see also J. C. Lagarias, Am. Math. Mon. 92, 3–23 (1985; Zbl 0566.10007)].
Benford’s law [F. Benford, Proc. Am. Philos. Soc. 78, 551–572 (1938; Zbl 0018.26502)] for an infinite sequence \(\{x_k \mid k\geq 1\}\) of positive quantities \(x_k\) is the assertion that \(\{\log x_k \mid k\geq 1\}\) is uniformly distributed \(\pmod 1\).
This paper studies the initial iterates \(x_k=T^{(k)}(x_0)\) for \(1\leq k\leq\mathbb{N}\) of the \(3x+1\) function, where \(N\) is fixed. It is shown in the main result of this paper (Theorem 2.1) that for most initial values \(x_0\), such sequences approximately satisfy Benford’s law, in the sense that the discrepancy of the finite sequence \(\{\log_Bx_k\mid k\geq 1\}\) is small; there \(B\geq 2\) is a fixed integer base.

MSC:
11B83 Special sequences and polynomials
11B37 Recurrences
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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