We find sufficient conditions for the singularity of a substitution $\mathbb{ Z}$-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution $\mathbb{ R}$-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector $\mathbf{ v}\in\mathbb{ Q}^d$, we find necessary and sufficient conditions for the orbit of $\omega\mathbf{ v}$ to be uniformly distributed modulo $1$ for almost every $\omega\in\mathbb{ R}$. We use our results to find new examples of singular substitution $\mathbb{ Z}$- and $\mathbb{ R}$-actions.
Categories: math.DS