Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density
Korec proved that, for each fixed θ > log 3/log 4, almost every positive integer n has a Collatz iterate below n^θ in natural density. Tao proved descent below every prescribed diverging threshold in logarithmic density. We obtain polylogarithmic descent in ordinary natural density within logarithmically many steps, with bounds on the preceding orbit values. The shortcut map T sends even n to n/2 and odd n to (3n+1)/2. The unaccelerated map Col sends even n to n/2 and odd n to 3n+1. For any prescribed positive function f(n) → ∞, however slowly, we prove that almost every positive integer n,…
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