Geometry, dynamics, and arithmetic of $S$-adic shifts

This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. Pure discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux-Rauzy as well as Brun substitutions (related to the respective continued fraction algorithms). It is shown that almost all these shifts have pure discrete spectrum, which proves a conjecture of Arnoux and Rauzy going back to the early nineties in a metric sense. We also prove that each linearly recurrent Arnoux-Rauzy shift with recurrent directive sequence has pure discrete spectrum. Using $S$-adic words related to Brun's continued fraction algorithm, we exhibit bounded remainder sets and natural codings for almost all translations on the two-dimensional torus.