Pisot substitution conjecture

Prove the Pisot substitution conjecture, namely, establish the conjectured pure discrete spectral behavior and associated toral-translation structure for fixed points of unimodular Pisot substitutions.

Background

The Pisot substitution conjecture is invoked as a major unresolved result underlying known transcendence consequences for fixed points of Pisot substitutions. In the paper's formulation, the conjecture would imply that fixed points of unimodular Pisot substitutions are measurably conjugate to translations on a finite-dimensional torus. The paper notes that the conjecture has been proved for substantial classes of substitutions but remains unresolved in general.