  Two conjectures are presented. The first, Conjecture 1, is that the
pushforward of a geometric distribution on the integers under $n$ Collatz
iterates, modulo $2^p$, is usefully close to uniform distribution on the
integers modulo $2^p$, if $p/n$ is small enough. Conjecture 2 is that the
density is bounded from zero for the incidence of both $0$ and $1$ for the
coefficients in the dyadic expansions of $-3^{-\ell }$ on all but an
exponentially small set of paths of a geometrically distributed random walk on
the two-dimensional array of these coefficients. It is shown that Conjecture 2
implies Conjecture 1. At present, Conjecture 2 is unresolved.
