  We prove that every infinite minimal subshift with word complexity $p(q)$
satisfying $\limsup p(q)/q < 3/2$ is measure-theoretically isomorphic to its
maximal equicontinuous factor; in particular, it has measurably discrete
spectrum. Among other applications, this provides a proof of Sarnak's
conjecture for all subshifts with $\limsup p(q)/q < 3/2$ (which can be thought
of as a much stronger version of zero entropy).
  As in \cite{creutzpavlov}, our main technique is proving that all
low-complexity minimal subshifts have a specific type of representation via a
sequence $\{\tau_k\}$ of substitutions, usually called an S-adic decomposition.
The maximal equicontinuous factor is the product of an odometer with a rotation
on a compact abelian connected one-dimensional group, for which we can give an
explicit description in terms of the substitutions $\tau_k$. We also prove that
all such odometers and groups may appear for minimal subshifts with $\limsup
p(q)/q = 1$, demonstrating that lower complexity thresholds do not further
restrict the possible structure.
