1. Definitions
The stopping time of \(n\) is the smallest positive integer \(s\) such that \(T^{s}(n)<n\). Terras observed that the density of integers whose stopping time exceeds any fixed bound tends to zero.
2. Tail Density
The probability of a parity walk remaining above the critical line of slope \(\log_2 3\) for all time is zero under the uniform measure, yet the corresponding set is nonempty in the 2-adic integers.
3. Implementation
Usage:
gcc -O2 -DLOGN=32 -o tailtest2 tailtest2.c && ./tailtest2