  It has been a long standing problem to find good symbolic codings for
translations on the $d$-dimensional torus that enjoy the beautiful properties
of Sturmian sequences like low factor complexity and good local discrepancy
properties. Inspired by Rauzy's approach we construct such codings in terms of
multidimensional continued fraction algorithms that are realized by sequences
of substitutions. In particular, given any exponentially convergent continued
fraction algorithm, these sequences lead to renormalization schemes which
produce symbolic codings of toral translations and bounded remainder sets at
all scales in a natural way.
  The exponential convergence properties of a continued fraction algorithm can
be viewed in terms of a Pisot type condition imposed on an attached symbolic
dynamical system. Using this fact, our approach provides a systematic way to
confirm purely discrete spectrum results for wide classes of symbolic dynamical
systems. Indeed, as our examples illustrate, we are able to confirm the Pisot
conjecture for many well-known families of sequences of substitutions. These
examples comprise classical algorithms like the Jacobi--Perron, Brun,
Cassaigne--Selmer, and Arnoux--Rauzy algorithms.
  As a consequence, we gain symbolic codings of almost all translations of the
$2$-dimensional torus having factor complexity $2n+1$ that are balanced for
words, which leads to multiscale bounded remainder sets. Using the Brun
algorithm, we also give symbolic codings of almost all $3$-dimensional toral
translations having multiscale bounded remainder sets.
