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Statistical \((3x+1)\) problem. (English) Zbl 1044.11006
The “Collatz”-problem (or “\(3x+1\)”- or “Hasse”- or “Syracuse”- or “Kakutani”-problem) is to prove that the sufficiently often iterated function \(T\) eventually comes to \(1\), where the function \(T(x)\) takes odd numbers \(x\) to \((3x+1)/2\) and even numbers \(x\) to \(x/2\). Let \(T^{(k)}\) denote the function if in the definition of \(T\) it is divided by \(2^k\), \(k>0\). Fixing \(k_1,k_2,\dots,k_m>0\) the author studies the set of odd integers to which one can apply \(T^{(k_1)}\), \(T^{(k_2)},\dots,T^{(k_m)}\) and in his Structure Theorem he determines this arithmetical progression explicitly. Using this result he proves some statistical statements which give some reason to expect that the \(3x+1\)-problem is true.

MSC:
11B37 Recurrences
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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References:
[1] Borovkov, Theory Probab Appl 45 pp 300– (2000)
[2] Lagarias, Amer Math Monthly 92 pp 3– (1985)
[3] The dynamical system generated by the 3n + 1 function. Lecture Notes in Mathematics, 1681. Springer-Verlag, Berlin, 1998. · Zbl 0892.11002
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