The lattice-avoidance criterion for the substitution property

Let $f$ be the characteristic function of a probability measure with a nontrivial absolutely continuous component, and let $\varphi$ be the density function of this component. Let $N_{\varphi}$ denote the set associated with $\varphi$ in the source, and let a lattice mean a set of the form $\tau+a\mathbb Z$ with $\tau\in\mathbb R$ and $a>0$. A characteristic function $f$ has the substitution property if, and only if, $N_{\varphi}$ does not contain any lattice $\tau+a\mathbb Z$, where $\tau\in\mathbb R$ and $a>0$. This criterion gives a precise characterization of when the substitution property holds for characteristic functions having an absolutely continuous component. The supplied text does not define $N_{\varphi}$ or identify whether this assertion is intended as a conjecture, theorem, or open problem, so its status remains unresolved here.