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# On the problem of positive predecessor density in $3n+1$ dynamics

• The $3n+1$ function is given by $T(n)=n/2$ for $n$ even, $T(n)=(3n+1)/2$ for $n$ odd. Given a positive integer $a$, another number $b$ is a called a predecessor of $a$ if some iterate $T^\nu(b)$ equals $a$. Here some ideas are described which may lead to a proof showing that the set of predecessors of $a$ has positive lower asymptotic density, for any positive integer $a\ne 0$ mod 3. Three unbridged gaps in the argument are formulated as conjectures.
Mathematics Subject Classification: 39A10, 11B85, 60J05.

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