Title: Noise Stability is computable and low dimensional,
Abstract: Questions of noise stability play an important role in hardness of
approximation in computer science as well as in the theory of voting. In many
applications, the goal is to find an optimizer of noise stability among all
possible partitions of $\mathbb{R}^n$ for $n \geq 1$ to $k$ parts with given
Gaussian measures $\mu_1,\ldots,\mu_k$. We call a partition $\epsilon$-optimal,
if its noise stability is optimal up to an additive $\epsilon$. In this paper,
we give an explicit, computable function $n(\epsilon)$ such that an
$\epsilon$-optimal partition exists in $\mathbb{R}^{n(\epsilon)}$. This result
has implications for the computability of certain problems in non-interactive
simulation, which are addressed in a subsequent work.