Bohr chaoticity of principal algebraic actions and Riesz product measures

For a continuous Nᵈ or Zᵈ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic Z actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of Zᵈ with positive entropy under the condition of existence of summable homoclinic points.