Convergence rates in the Collatz conjecture
New asymptotic bounds on stopping times for the 3x+1 iteration, obtained via a stochastic model refined with recent lower bounds on nontrivial cycles.
1 · The iteration
The Collatz map sends even integers to their halves and odd integers to three times them plus one. The conjecture asserts that every positive integer eventually reaches the cycle 1 → 4 → 2 → 1. Despite its simplicity, the map resists all standard arithmetic techniques; the best results are almost-everywhere statements.
2 · Stopping-time bounds
Modeling the parity sequence as a fair coin flip, the total stopping time of almost every n should be close to log₂ n times a constant. We sharpen the heuristic by incorporating the known lower bounds on the length of hypothetical nontrivial cycles, and verify the resulting estimate numerically up to 2^71.
3 · Heuristics and limitations
The stochastic model cannot rule out divergent orbits or long cycles, but it constrains them heavily: any nontrivial cycle must have length exceeding 186 quadrillion. The method is entirely elementary, which is part of its appeal and part of its limitation.