  Van der Corput and Halton sequences are well-known low-discrepancy sequences.
Almost twenty years ago Ninomiya defined analogues of van der Corput sequences
for $\beta$-numeration and proved that they also form low-discrepancy sequences
if $\beta$ is a Pisot number. Only very recently Robert Tichy and his
co-authors succeeded in proving that $\boldsymbol{\beta}$-adic Halton sequences
are equidistributed for certain parameters
$\boldsymbol{\beta}=(\beta_1,\ldots,\beta_s)$ using methods from ergodic
theory. In the present paper we continue this research and give discrepancy
estimates for $\boldsymbol{\beta}$-adic Halton sequences for which the
components $\beta_i$ are $m$-bonacci numbers. Our methods are quite different
and use dynamical and geometric properties of Rauzy fractals that allow to
relate $\boldsymbol{\beta}$-adic Halton sequences to rotations on high
dimensional tori. The discrepancies of these rotations can then be estimated by
classical methods relying on W.~M.~Schmidt's Subspace Theorem.
