Difference functions of periodic measurable functions

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 $Δ_h f(x)=f(x+h)-f(x)$ 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, $ℌ(ℱ,G) = {H ⊂ ℝ/ℤ : (∃f ∈ ℱ \ G) (∀ h ∈ H) Δ_h f ∈ G}$, 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 $\mathbb{T}=ℝ/ℤ$ that are invariant for changes on null-sets (e.g. measurable functions, $L_p$, $L_∞$, essentially continuous functions, functions with absolute convergent Fourier series (ACF*), essentially Lipschitz functions) and classes of continuous functions on $\mathbb{T}$ (e.g. continuous functions, continuous functions with absolute convergent Fourier series, Lipschitz functions). The classes ℌ(ℱ,G) are often related to some classes of thin sets in harmonic analysis (e.g. $ℌ(L_1,{ACF}*)$ is the class of N-sets). Some results concerning the difference property and the weak difference property of these classes of functions are also obtained.