  As indicated by the title, this paper investigates three closely related
topics -- R\'enyi resolvability, noise stability, and anti-contractivity. The
R\'enyi resolvability problem refers to approximating a target output
distribution of a given channel in the R\'enyi divergence when the input is set
to a function of a given uniform random variable. This problem for the R\'enyi
parameter in $[0,2]\cup\{\infty\}$ was studied by the present author and Tan in
2019. In the present paper, we provide a complete solution to this problem for
the R\'enyi parameter in the entire range $\mathbb{R}\cup\{\pm\infty\}$. We
then connect the R\'enyi resolvability problem to the noise stability problem,
by observing that the $q$-stability of a set can be expressed in terms of the
R\'enyi divergence between the true output distribution and the target
distribution in a variant of the R\'enyi resolvability problem. By such a
connection, we provide sharp dimension-free bounds on the $q$-stability. We
lastly relate the noise stability problem to the anti-contractivity of a Markov
operator (i.e., conditional expectation operator), where anti-contractivity
introduced by us refers to as the opposite property of the well-known
contractivity/hyercontractivity. We derive sharp dimension-free
anti-contractivity inequalities. All of the results in this paper are evaluated
for binary distributions. Our proofs in this paper are mainly based on the
method of types, especially strengthened versions of packing-covering lemmas.
