Sinai, Ya. G. Statistical \((3x+1)\) problem. (English) Zbl 1044.11006 Commun. Pure Appl. Math. 56, No. 7, 1016-1028 (2003). 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. Reviewer: Helmut Müller (Hamburg) Cited in 3 Documents MSC: 11B37 Recurrences 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) Keywords:\(3x+1\)-problem; Wiener trajectories PDF BibTeX XML Cite \textit{Ya. G. Sinai}, Commun. Pure Appl. Math. 56, No. 7, 1016--1028 (2003; Zbl 1044.11006) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.