On minimal subshifts of linear word complexity with slope less than 3/2

We prove that every infinite minimal subshift with word complexity $p(q)$ satisfying $\mathrm{lim\hspace{0.17em}sup}p(q)\mathrm{/}q<3\mathrm{/}2$ 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 $\mathrm{lim\hspace{0.17em}sup}p(q)\mathrm{/}q<3\mathrm{/}2$ (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 $\mathrm{lim\hspace{0.17em}sup}p(q)\mathrm{/}q=1$, demonstrating that lower complexity thresholds do not further restrict the possible structure.