We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions ${\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ(ℱ,G)=H\subset ℝ/\mathbb{Z}:\left(\exists f\in ℱG\right)\left(\forall h\in H\right){\Delta }_{h}f\in G$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $𝕋=ℝ/\mathbb{Z}$ that are invariant for changes on null-sets (e.g. measurable...