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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)
##### Keywords:
$$3x+1$$-problem; Wiener trajectories
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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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