Density Bounds for the 3x+1 Problem II: Krasikov Inequalities

The 3x+1 function T(x) = (3x+1) over 2 if x is odd and x over 2 if x is even. Let pi sub a (x) count the number of n with |n| = x whic eventually reach a under iteration by T. Then for any a not= O (mod 3) and sufficiently large x, pi sub a (x) > = x sup (.804). The proof is based on solving nonlinear programming problems constructed from inequalities of Krasikov