  We consider one-dimensional substitution tiling spaces where the dilatation
(stretching factor) is a degree d Pisot number, and where the first rational
Cech cohomology is d-dimensional. We construct examples of such "homological
Pisot" substitutions that do not have pure discrete spectra. These examples are
not unimodular, and we conjecture that the coincidence rank must always divide
a power of the norm of the dilatation. To support this conjecture, we show that
homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in
which the number of occurrences of a patch for a return length is governed
strictly by the length. The ERP puts strong constraints on the measure of any
cylinder set in the corresponding tiling space.
